...lly in the meaning of [[Von Neumann algebra|von Neumann algebra]] or $W^*$-algebra.
2) A [[Ring with operators|ring with operators]].
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...mple of an non-associative commutative ring is a Jordan ring (see [[Jordan algebra]]).
[[Category:Algebra]]
293 bytes (40 words) - 20:57, 2 November 2014
...e algebra over a field possessing a unit element is imbeddable in a simple algebra with the same unit element.
For references, see [[Simple ring]].
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...ar-algebra(2)|linear algebra]], [[multilinear algebra]], and [[topological algebra]].
An [[algebraic system]], consisting of a set with a family of algebraic operations.
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The centre of a ring $R$ is
the collection $Z$ of all elements of the ring $R$ that commute with every
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''unitary ring, ring with identity, ring-with-a-one''
...ltiplication $(m,r) \cdot (n,s) = (mn, ms + nr + rs)$ which has $(1,0)$ as a multiplicative identity.
1 KB (199 words) - 18:38, 13 November 2023
...erfect]]. A [[Noetherian ring|Noetherian]] balanced ring is an [[Artinian ring]].
<TR><TD valign="top">[1]</TD> <TD valign="top"> ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''19''' (1981) pp. 31–134</TD></TR>
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...ing|ring]] which is a left (right) [[Semi-chain module|semi-chain module]] over itself.
...e of a serial ring is the ring of formal power series in one variable over a field.
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...trass rings are [[analytic ring]]s (rings of convergent power series) over a perfect field, to which the [[Weierstrass preparation theorem]] is applicab
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''with respect to a radical $r$''
...ebras]]; [[Jordan algebra|Jordan algebra]]; [[Lie algebra, semi-simple|Lie algebra, semi-simple]]).
1,013 bytes (144 words) - 16:31, 5 August 2014
A ring with division is a (not necessarily associative) [[ring]] in which the equations
$$a \cdot x = b \quad;\quad\quad y \cdot a = b $$
1 KB (170 words) - 17:50, 28 December 2013
...operator ring]], when the domain of operators is a commutative associative ring with unit.
...onal vector spaces and linear operators between them. See the entry Linear algebra below.
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...acobson ring. On the contrary, a local non-Artinian ring is not a Jacobson ring.
...o the spectrum $\mathrm{Spec}(A)$; this definition leads to the concept of a Jacobson scheme.
3 KB (415 words) - 20:32, 19 January 2016
...full matrix ring of a [[local ring]]. This holds, in particular, if $J$ is a [[nil ideal]].
<TR><TD valign="top">[2]</TD> <TD valign="top"> C. Faith, "Algebra" , '''1–2''' , Springer (1973–1976)</TD></TR>
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...e additive group of $B$ and for which $(ab)\phi=b\phi\cdot a\phi$ ($a,b\in A$).
The [[opposite ring]] $R^{\mathrm{op}}$ to a ring $R$ is anti-isomorphic to $R$.
624 bytes (103 words) - 19:11, 24 November 2023
...e. $\partial(\alpha x) = \alpha \partial(x)$, where $\alpha \in K$, $x \in A$.
Cf. also [[Derivation in a ring]].
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''$M$ over a commutative ring $R$''
.... For example, for a finite Abelian group$M$ regarded as a module over the ring of integers, $\mathrm{Supp}(M)$ consists of all prime ideals $(p)$, where $
918 bytes (149 words) - 20:46, 18 October 2014
The term ''regular algebra'' may refer to:
* a [[regular ring (in commutative algebra)]];
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...he first one is. Every automorphism of a finite-dimensional central simple algebra is inner and its dimension is the square of an integer.
...="top">[2]</TD> <TD valign="top"> L.A. Skornyakov, "Elements of general algebra" , Moscow (1983) (In Russian)</TD></TR>
2 KB (250 words) - 14:32, 7 April 2023
...o be strongly degenerate. The smallest ideal of a Lie algebra the quotient algebra by which is not strongly degenerate is called the Kostrikin radical.
...60 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.I. Kostrikin, "Around Burnside" , Springer (1989) (Translated from Russ
1 KB (218 words) - 16:32, 1 May 2014
''integral ring''
..., is an integral domain. Conversely, an integral domain can be imbedded in a field. Such an imbedding is given by the construction of the [[field of fra
2 KB (265 words) - 16:14, 11 September 2016
...s, all elements of an Engel algebra are [[Engel element]]s, see also [[Lie algebra, nil]]).
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''right Artinian ring''
...mber $m$ such that $B_m=B_{m+1}=\dotsb$. The definition of a left Artinian ring is similar.
3 KB (460 words) - 12:07, 14 February 2020
The same as a normed or [[Banach algebra|Banach algebra]].
A ring with a [[Valuation|valuation]] on it.
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...simple rings, cf. also [[Noetherian ring|Noetherian ring]]) different from a skew-field, as well as Noetherian simple rings with zero divisors but witho
...gebras|Alternative rings and algebras]]). See also [[Simple algebra|Simple algebra]].
2 KB (318 words) - 09:55, 26 November 2016
...an ring $K\neq0$ is a direct sum of fields $\mathbf Z_2$ and therefore has a unit element.
...ebra|Boolean algebra]], namely: Any Boolean algebra is a Boolean ring with a unit element under the operations of addition and multiplication defined by
2 KB (335 words) - 15:09, 13 August 2014
...$k$ is a perfect field, an analytic ring is an [[Excellent ring|excellent ring]].
...ntiels de régularité pour les localisés des algèbres analytiques" ''J. of Algebra'' , '''5''' (1967) pp. 305–324</TD></TR><TR><TD valign="top">[2]</TD> <
2 KB (341 words) - 12:10, 25 August 2014
...that a non-null subalgebra of a free algebra over a field $\Phi$ is a free algebra.
* Kurosh, A. "Nonassociative free algebras and free products of algebras". Mat. Sb., N
1 KB (163 words) - 08:20, 12 November 2023
$#A+1 = 44 n = 0
$#C+1 = 44 : ~/encyclopedia/old_files/data/R082/R.0802450 Ring with operators,
4 KB (634 words) - 07:02, 30 March 2024
...hat R)$ is the endomorphism ring of the right $R$-module $\widehat R$. The ring $Q_{\mathrm{max}}(R)$ can also be defined as the [[direct limit]]
...f all dense right ideals of $R$ (a right ideal $D$ of a ring $R$ is called a dense ideal if
2 KB (291 words) - 07:26, 29 November 2014
====Separable algebra over a field====
...$ is separable if and only if the centres of the simple components of this algebra (see [[Associative rings and algebras|Associative rings and algebras]]) are
2 KB (316 words) - 21:57, 29 November 2014
...[2]]] established that any [[Noetherian ring|Noetherian ring]] is a Lasker ring.
...D> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR></table>
1 KB (157 words) - 11:43, 29 June 2014
...irect sum of modules dual to finitely-generated right ideals of the ground ring.
...are projective. The ring of polynomials in one variable over a commutative ring $T$ is semi-hereditary if and only if $R$ is regular in the sense of von Ne
3 KB (403 words) - 17:47, 27 August 2014
...of finite type over a Bezout ring is a direct sum of a torsion module and a free module.
1 KB (190 words) - 19:52, 2 November 2014
...lar, associative) ring $R$ is called quasi-regular if there is an element $a'\in R$ such that
$$a+a'+aa'=a+a'+a'a=0.$$
1 KB (189 words) - 21:27, 6 September 2017
...= R$, then $R$ is said to be integrally closed in $S$ (cf. also [[Integral ring]]).
...nd is closed in its field of fractions. In some of the literature a normal ring is also required to be an integral domain.
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''of a ring $R$''
The ring $R^{\mathrm{op}}$ having the same additive group as $R$ but with multiplica
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''over a field $F$''
...over a field $F$ that generalises the construction of the [[quaternion]]s over the field of real numbers.
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...om the properties of the category of all left (or right) modules over this ring (cf. [[Morita equivalence]]; [[Module]]).
...their projectivity, and also to the injectivity of all left ideals of the ring [[#References|[1]]].
2 KB (327 words) - 07:32, 13 December 2016
...determines a [[closed subscheme]] of the [[affine scheme]] $\mathrm{Spec}\,A$, consisting of the points that are not normal.
...D> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR>
1 KB (161 words) - 08:20, 30 November 2014
...ing has a left classical ring of fractions that is a classical semi-simple ring if and only if it satisfies the maximum condition for left annihilators and
952 bytes (138 words) - 19:44, 19 October 2014
A collection $\mathcal{A}$ of subsets of a set $X$ satisfying:
i) $\emptyset\in \mathcal{A}$;
2 KB (231 words) - 09:47, 16 August 2013
...r matrices over a field the matrices with 0's along the main diagonal form a nilpotent ideal.
...in a [[Noetherian ring|Noetherian ring]]. In a left (or right) Noetherian ring every left (right) nil ideal is nilpotent.
3 KB (470 words) - 23:58, 24 November 2018
...es the ring of integers and all complete local rings. See also [[Excellent ring]].
...="top">[1]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975)</TD></TR>
2 KB (317 words) - 17:54, 10 January 2016
...tive ring $R$ is said to be non-full if it can be represented in the form $A=BC$ where $B$, $C$ are matrices of orders $n\times r$ and $r\times n$, resp
$$A=(a,a_2,\ldots,a_n),\quad B=(b,a_2,\ldots,a_n)$$
5 KB (851 words) - 14:57, 30 July 2014
...and algebras]]). Right perfect rings are defined similarly. A left perfect ring need not be right perfect.
...-module has a decomposition with respect to which every direct summand has a complement (see [[Krull–Remak–Schmidt theorem|Krull–Remak–Schmidt t
3 KB (491 words) - 19:59, 30 October 2016
''(left), left $RR$-ring''
...visors. A ring of polynomials over a commutative Rickart ring is a Rickart ring.
2 KB (418 words) - 06:38, 18 October 2017
$#A+1 = 6 n = 2
...ive module]]). The Prüfer domains are the commutative integral domains $ A $
2 KB (297 words) - 08:08, 6 June 2020
$#A+1 = 30 n = 0
...~/encyclopedia/old_files/data/R080/R.0800820 Regular ring (in commutative algebra)
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''of a ring $R$ by an ideal $I$''
$$(a+I)(b+I)=ab+I.$$
2 KB (270 words) - 14:56, 30 December 2018
...m was first obtained by J. Wedderburn for finite-dimensional algebras over a field, and was proved by E. Artin [[#References|[1]]] in its final formulat
...Artin, "The influence of J.H.M. Wedderburn on the development of modern algebra" ''Bull. Amer. Math. Soc.'' , '''56''' (1950) pp. 65–72</TD></TR>
1 KB (211 words) - 06:17, 13 September 2016
A topological ring is a [[Ring|ring]] $R$ that is a [[Topological space|topological space]], and such that the mappings
...gical ring. A direct product of topological rings is a topological ring in a natural way.
3 KB (410 words) - 21:45, 23 July 2012
$#A+1 = 15 n = 0
$#C+1 = 15 : ~/encyclopedia/old_files/data/K055/K.0505930 Krull ring,
3 KB (424 words) - 22:15, 5 June 2020
$#A+1 = 47 n = 0
$#C+1 = 47 : ~/encyclopedia/old_files/data/E036/E.0306760 Excellent ring
4 KB (643 words) - 19:38, 5 June 2020
$#A+1 = 29 n = 0
$#C+1 = 29 : ~/encyclopedia/old_files/data/S083/S.0803900 Self\AAhinjective ring,
4 KB (550 words) - 08:13, 6 June 2020
$#A+1 = 57 n = 0
$#C+1 = 57 : ~/encyclopedia/old_files/data/H046/H.0406940 Hensel ring
5 KB (787 words) - 22:10, 5 June 2020
An ideal in a ring or algebra that is not a [[Prime ideal|prime ideal]].
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$#A+1 = 36 n = 0
''of a ring''
5 KB (764 words) - 08:09, 6 June 2020
$#A+1 = 37 n = 0
$#C+1 = 37 : ~/encyclopedia/old_files/data/P072/P.0702750 Place of a field
2 KB (326 words) - 08:06, 6 June 2020
...ain modules if and only if $R$ is a generalized [[Uniserial ring|uniserial ring]].
...a$-uniserial are used for a direct sum of uniserial modules (respectively, a finite direct sum of uniserial modules).
751 bytes (111 words) - 16:40, 1 May 2014
$#A+1 = 27 n = 0
$#C+1 = 27 : ~/encyclopedia/old_files/data/N066/N.0606850 Noetherian ring,
3 KB (539 words) - 18:51, 3 April 2024
''unique factorisation domain'', ''Gaussian ring''
...irreducible elements $P$ such that every non-zero element $a\in A$ admits a unique representation
3 KB (480 words) - 21:45, 3 January 2021
...ising the properties of a [[Lie algebra]]. A Leibniz algebra $L$ is a $K$-algebra with multiplication denoted by $[\cdot,\cdot]$ satisfying
Every [[Lie algebra]] is a Leibniz algebra, and a Leibniz algebra is a Lie algebra if in addition $[x,x] = 0$.
1 KB (217 words) - 15:05, 19 November 2023
$#A+1 = 9 n = 0
$#C+1 = 9 : ~/encyclopedia/old_files/data/S091/S.0901020 Subrepresentation of a representation
812 bytes (117 words) - 08:24, 6 June 2020
A non-zero [[unital module]] $M$ over a [[unital ring]] $R$ that contains only two submodules: the null module and $M$ itself.
...]]s of $G$ over $K$ are precisely the irreducible modules over the [[group algebra]] $R = k[G]$.
2 KB (391 words) - 18:11, 18 March 2018
$#A+1 = 21 n = 0
A left (or right) module $ M $
2 KB (372 words) - 19:39, 5 June 2020
...algebraic operations on tensors (cf. [[Tensor on a vector space|Tensor on a vector space]]) are studied.
...r a commutative associative ring $A$ with unit is the algebra $T(V)$ over $A$ whose underlying module has the form
2 KB (329 words) - 04:49, 23 July 2018
$#A+1 = 33 n = 0
$#C+1 = 33 : ~/encyclopedia/old_files/data/D030/D.0300550 Dedekind ring
4 KB (637 words) - 17:32, 5 June 2020
$#A+1 = 18 n = 0
A generalization of the notion of an associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]). Le
3 KB (408 words) - 08:11, 6 June 2020
''of a closed symmetric algebra $R$ of bounded linear operators on a Hilbert space $H$''
A commutative symmetric Banach algebra $E$ of operators on $H$ such that $(R\cup E)'=E$. Diagonal algebras are emp
2 KB (264 words) - 11:41, 26 March 2023
...X$ is Noetherian if and only if every open subset of $X$ is quasi-compact. A Noetherian space is the union of finitely many irreducible components.
...[[Noetherian ring|Noetherian ring]], where $J$ is the [[nil radical]] of $A$.
1 KB (193 words) - 17:12, 29 December 2020
$#A+1 = 29 n = 0
A [[Fractional ideal|fractional ideal]] $ \mathfrak a $
3 KB (406 words) - 21:43, 3 January 2021
...athbf{C} _ { Q } ( R )$. If $R$ is semi-prime (cf. also [[Prime ring|Prime ring]]), the definition is similar, but considerably more complicated.
...rime ring satisfying a non-trivial differential identity must also satisfy a non-trivial generalized identity.
4 KB (655 words) - 16:46, 1 July 2020
''in a class $\mathfrak K$ of universal algebras''
...(Lie algebras, lattices, Boolean algebras), and sometimes it does not have a recursive solution (alternative rings).
2 KB (357 words) - 13:01, 9 April 2014
''on a ring $A$''
...modules. The most extensively used is the [[Adic topology|adic topology]], a basis of which is given by the powers of an ideal.
2 KB (315 words) - 19:21, 14 August 2014
A left (or right) [[module]] $M$ over a [[ring with identity]] $e$ such that multiplication by $e$ is the identity operato
...fusion in that it may suggest some module generalization of the concept of a unitary vector space, cf. [[Unitary space]].
1 KB (178 words) - 16:19, 11 September 2016
''of a commutative ring $R$''
...the Artin–Rees condition: For arbitrary left ideals $I,J$ of $R$ there is a natural number $n$ such that $I^n \cap J \subseteq IJ$ (cf. [[#References|[
3 KB (455 words) - 19:51, 5 October 2017
...f free generators into an arbitrary ring of the variety can be extended to a homomorphism.
...cteristic zero and if $M_2\not\subseteq\mathfrak M$, then $\mathfrak M$ is a Specht variety.
4 KB (701 words) - 17:01, 23 November 2023
$#A+1 = 73 n = 0
...73 : ~/encyclopedia/old_files/data/I051/I.0501450 Integral extension of a ring
3 KB (562 words) - 22:12, 5 June 2020
...is both Artinian and [[Noetherian module|Noetherian]]. See also [[Artinian ring]].
|valign="top"|{{Ref|Fa}}||valign="top"| C. Faith, "Algebra: rings, modules, and categories" , '''1''' , Springer (1973)
948 bytes (141 words) - 19:06, 25 October 2014
...e situation ${}_R M_E$, where $E$ is the ring of endomorphisms of $M$. Any ring $R$ can be given the natural structure of an $(R,R)$-bimodule.
...t $S$-linear. The category of $(R,S)$-bimodules with bimodule morphisms is a [[Grothendieck category]].
2 KB (277 words) - 17:01, 23 November 2023
''group algebra of a group $G$ over a field $K$''
...a]] over $K$ whose elements are all possible finite sums of the type $\def\a{\alpha}\def\b{\beta}\sum_{g\in G}\a_g g$, $g\in G$, $\a_g\in K$, the operat
5 KB (872 words) - 22:00, 29 April 2012
''full matrix ring''
...cle $R$ is an [[Associative rings and algebras|associative ring]] [[unital ring|with identity]].
4 KB (631 words) - 06:20, 13 September 2016
''Stanley–Reisner face ring, face ring''
...of a [[simplicial complex]] $\Delta$ over a [[field]] $k$ is the quotient ring
3 KB (544 words) - 20:16, 8 November 2023
...sets. Given two sets $A$ and $B$, their symmetric difference, denoted by $A \Delta B$, is given by
...\cup (B \setminus A) = (A \cup B) \setminus (A \cap B) = (A \cap B') \cup (A' \cap B)
2 KB (273 words) - 08:47, 29 April 2023
$#A+1 = 91 n = 0
be a [[Commutative ring|commutative ring]] with unit element and $ \underline{x} = ( x _ {1} \dots x _ {r} ) $
6 KB (934 words) - 22:15, 5 June 2020
$#A+1 = 20 n = 0
A statement obtained by K. Hensel [[#References|[1]]] in the creation of the
3 KB (386 words) - 22:10, 5 June 2020
$#A+1 = 63 n = 0
$#C+1 = 63 : ~/encyclopedia/old_files/data/E035/E.0305610 Endomorphism ring
4 KB (595 words) - 19:37, 5 June 2020
$#A+1 = 40 n = 0
An element $ a $
3 KB (489 words) - 02:30, 11 July 2022
...
a topological vector space, then
a linear representation of $X$ on $E$ is
a representation whose image contains only continuous linear operators on $E$
<table><TR><TD valign="top">[1]</TD> <TD valign="top">
A.
A. Kirillov, "Elements of the theory of representations" , Springer (1976)
1 KB (197 words) - 19:18, 25 October 2014
...ts elements is bounded. Subalgebras and homomorphic images of an algebraic algebra (of bounded degree) are algebraic algebras (of bounded degree).
...ebra]]s and associative [[skew-field]]s with a countable set of generators over an uncountable field.
2 KB (305 words) - 16:55, 16 October 2014
$#A+1 = 16 n = 0
$#C+1 = 16 : ~/encyclopedia/old_files/data/Z099/Z.0909270 Zhegalkin algebra
2 KB (316 words) - 08:29, 6 June 2020
$#A+1 = 98 n = 0
$#C+1 = 98 : ~/encyclopedia/old_files/data/D031/D.0301210 Derivation in a ring
6 KB (945 words) - 17:32, 5 June 2020
$#A+1 = 56 n = 0
$#C+1 = 56 : ~/encyclopedia/old_files/data/A010/A.0100790 Adic topology
3 KB (559 words) - 16:09, 1 April 2020
$#A+1 = 48 n = 0
$#C+1 = 48 : ~/encyclopedia/old_files/data/G044/G.0404660 Graded algebra
4 KB (576 words) - 19:42, 5 June 2020
...3]]]). The isolated prime ideals of an ideal $I$ of a polynomial ring over a field correspond to the irreducible components of the [[affine variety]] of
...lign="top"> M.F. Atiyah, I.G. Macdonald, "Introduction to commutative algebra" , Addison-Wesley (1969)</TD></TR>
2 KB (386 words) - 20:06, 5 October 2017
...ny $r \in R$ and $m \in M$, is surjective. A module $P$ over a ring $R$ is a generator of the category of $R$-modules if and only if $P$ is balanced as
<TR><TD valign="top">[1]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules and categories" , '''1–2''' , Springer (1973–1976)</TD
646 bytes (112 words) - 20:23, 10 October 2017
$#A+1 = 37 n = 0
...: ~/encyclopedia/old_files/data/E037/E.0307030 Extension of an associative algebra
2 KB (359 words) - 19:38, 5 June 2020
...roup $F_4$ is the group of automorphisms of the exceptional Jordan algebra over $\mathbf C$.
...55</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.B. Vinberg, "A construction of the exponential simple Lie algebras" ''Trudy Sem. Vektor. T
3 KB (477 words) - 19:58, 15 March 2023
$#A+1 = 35 n = 1
whose objects are the right unitary modules over an arbitrary associative ring $ R $
4 KB (572 words) - 08:01, 6 June 2020
$#A+1 = 33 n = 0
The simplest form of an algebraic expression, a [[Polynomial|polynomial]] containing only one term.
2 KB (376 words) - 08:01, 6 June 2020
...mod{\mathfrak a_i}$, $i=1,\dots,n$. In the particular case when $A$ is the ring of integers $\mathbf Z$, the Chinese remainder theorem states that for any
...an $x\in A$ such that the decomposition of the principal ideal $(x)$ into a product of prime ideals has the form
2 KB (355 words) - 13:31, 11 July 2018
A ''skew-field'' (or ''skew field'') is a
...$ya=b$ with $a\ne 0$ are uniquely solvable. In the case of an associative ring (cf.
4 KB (607 words) - 19:53, 5 March 2012
...only the target of $\mu$ is a topological vector space $V$ (more generally a commutative topological group) or the extended real line $[-\infty, \infty]
...of definition is a [[Ring of sets|ring]] (more often an [[Algebra of sets|algebra]]) and $\mu$ has the property that
3 KB (459 words) - 14:45, 19 September 2012
$#A+1 = 14 n = 0
A [[Scheme|scheme]] $ ( X , {\mathcal O} _ {X} ) $
3 KB (393 words) - 08:10, 6 June 2020
$#A+1 = 81 n = 0
$#C+1 = 81 : ~/encyclopedia/old_files/data/P074/P.0704740 Principal ideal ring
5 KB (880 words) - 19:00, 9 January 2024
''in a class $\mathfrak K$ of universal algebras $A_\alpha$, $\alpha\in\Omega$, fr
...direct sum. In certain cases there is a description of the subalgebras of a free product, for example, in groups (see [[Free product of groups|Free pro
2 KB (259 words) - 20:56, 30 July 2014
$#A+1 = 19 n = 0
$#C+1 = 19 : ~/encyclopedia/old_files/data/C023/C.0203010 Coherent ring
3 KB (479 words) - 17:45, 4 June 2020
''polynomial ring''
...er $R$; the set of variables serves as a system of free generators of this algebra.
6 KB (1,011 words) - 06:01, 30 September 2023
''in commutative algebra''
...lent to the assertion "every point $\mathfrak{P}$ of $\text{Spec}\,A$ has a neighbourhood in which $P$ holds" .
2 KB (320 words) - 08:07, 1 January 2017
''crossed product, of a [[group]] $G$ and a [[ring]] $K$''
''of a vector $a$ by a vector $b$ in $\mathbb{R}^3$, see [[Vector product]].''
4 KB (702 words) - 08:53, 2 June 2016
...mutative ring]] $R$ with unit element in the commuting variables from $X$. A polynomial or power series $f(X) \in R[[X]]$ is called ''symmetric'' if for
...and descent sets of permutations [[#References|[a3]]]. This time one takes a totally ordered set of indeterminates, e.g. $V = \{V_1,V_2,\ldots\}$, with
4 KB (708 words) - 13:51, 20 March 2023
...]] of its field of coefficients. The absolutely irreducible polynomials of a single variable are the polynomials of degree 1. In the case of several var
...f arbitrarily high degree; for example, $x^n+px+p$, where $n>1$ and $p$ is a prime number, is irreducible in $\mathbf Q[x]$ by Eisenstein's criterion (s
3 KB (478 words) - 15:26, 30 December 2018
$#A+1 = 72 n = 0
...~/encyclopedia/old_files/data/L060/L.0600280 Localization in a commutative algebra
4 KB (713 words) - 21:35, 4 January 2021
''of a [[graded module]] $M = \bigoplus_n M_n$''
...the ''Hilbert function'' of $M$ is $h_M(n) = \dim_K M_n$ and there exists a polynomial $P_M(t)$ with rational coefficients such that, for sufficiently
2 KB (361 words) - 05:51, 17 April 2024
$#A+1 = 34 n = 0
...all values of which can be expressed by one (real) number. More generally, a scalar is an element of some [[Field|field]].
2 KB (358 words) - 08:12, 6 June 2020
''left, of a set $X$ in $R$''
...poid) with a zero. The right annihilator of a set $X$ in $R$ is defined in a similar manner as the set
2 KB (337 words) - 17:14, 15 November 2023
$#A+1 = 27 n = 0
$#C+1 = 27 : ~/encyclopedia/old_files/data/C023/C.0203160 Cohomology ring
3 KB (405 words) - 18:45, 13 January 2024
...ages/c/c025/c025860/c02586012.png" /> is
a continuous representation, then
a contraction of <img align="absmiddle" border="0" src="https://www.encyclope
...//www.encyclopediaofmath.org/legacyimages/c/c025/c025860/c02586055.png" />
over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
8 KB (1,091 words) - 17:21, 7 February 2011
...mes x$ ($x,y \in M$). A symmetric algebra is a commutative associative $A$-algebra with an identity. It is graded:
...omorphism of $S(M)$ onto the polynomial algebra $A[X_1,\ldots,X_n]$ (see [[Ring of polynomials]]).
3 KB (487 words) - 18:21, 11 April 2017
$#A+1 = 122 n = 0
$#C+1 = 122 : ~/encyclopedia/old_files/data/D110/D.1100090 Defect group of a block
6 KB (919 words) - 17:32, 5 June 2020
$#A+1 = 62 n = 0
...62 : ~/encyclopedia/old_files/data/D032/D.0302260 Differential operator on a module
5 KB (785 words) - 19:35, 5 June 2020
$#A+1 = 24 n = 0
A two-sided [[Ideal|ideal]] $ P $
2 KB (393 words) - 08:07, 6 June 2020
$#A+1 = 74 n = 0
$#C+1 = 74 : ~/encyclopedia/old_files/data/C022/C.0202970 Cohen\ANDMacaulay ring,
7 KB (1,158 words) - 17:45, 4 June 2020
...morphism is normal. The concept of a normal epimorphism is dual to that of a [[normal monomorphism]].
[[Category:Category theory; homological algebra]]
941 bytes (147 words) - 21:29, 1 November 2014
...inear representation]] $\pi$ of a group (ring, algebra, semi-group) $X$ in a vector space $E$ such that if $\pi(x)\xi=0$ for some $\xi\in E$ and all $x\
212 bytes (38 words) - 12:59, 19 April 2014
$#A+1 = 18 n = 0
A [[module]] $M$ over a [[commutative ring]] $A$
2 KB (308 words) - 18:37, 11 April 2023
A non-empty set $S$ with two associative binary operations $+$ and $\cdot$, s
(a+b) \cdot c = a\cdot c + b \cdot c
2 KB (371 words) - 05:54, 15 April 2023
...t possible to test for projectivity (injectivity) of
a module by computing
a single <img align="absmiddle" border="0" src="https://www.encyclopediaofmat
...r any right-perfect
ring (cf. also [[Perfect
ring|Perfect
ring]]) there is
a proper class of p-test modules. If $R$ is not right perfect, then it is con
5 KB (732 words) - 17:46, 1 July 2020
$#A+1 = 11 n = 0
A subset $ A $
1 KB (240 words) - 08:02, 6 June 2020
''of a module $M$ over a ring $R$''
...ly-reducible module]]). If $\phi$ is a homomorphism from a module $A$ into a module $B$, then the set
2 KB (361 words) - 19:20, 2 October 2016
$#A+1 = 17 n = 0
of a prime ideal $ \mathfrak p $
3 KB (456 words) - 17:43, 11 January 2021
$#A+1 = 41 n = 0
$#C+1 = 41 : ~/encyclopedia/old_files/data/A011/A.0101160 Affine variety,
3 KB (467 words) - 20:48, 15 March 2023
A field $k$ is algebraically closed if any polynomial of non-zero degree over $k$ has at
field $k$ each polynomial of degree $n$ over $k$ has exactly $n$ roots
1 KB (201 words) - 21:31, 5 March 2012
...$R$ be a finitely-generated $k$-algebra, that is, a homomorphic image of a ring of polynomials $R = k [ x _ { 1 } , \dots , x _ { n } ] / I$.
A $k$-derivation of $R$ is a $k$-linear mapping $\delta : R \rightarrow R$ that satisfies the Leibniz ru
5 KB (705 words) - 16:52, 1 July 2020
...Z}\langle x_1,x_2,\ldots\rangle$ which vanishes under all substitutions of ring elements for the variables. The ''standard polynomial'' of degree $n$ is th
...ki theorem says that the ring of $(n\times n)$-matrices over a commutative ring satisfies the standard polynomial of degree $2n$, and no polynomial of lowe
4 KB (628 words) - 21:20, 11 December 2017
...in which an associative multiplication and addition exist; a near-ring is a (not necessarily Abelian) group with respect to addition, and the right dis
must hold too. A near-ring is also an example of a [[Multi-operator group|multi-operator group]].
5 KB (767 words) - 11:23, 27 October 2014
...derives from the "interpretation" of $(A,B)$ as (the essential data of) a discrete-time time-invariant linear control system:
...\in R^n$, $u(t) \in R^m$, or also, when $R = \mathbf{R}$ or $\mathbf{C}$, a continuous-time time-invariant linear control system:
5 KB (799 words) - 20:38, 12 November 2017
$#A+1 = 33 n = 0
$#C+1 = 33 : ~/encyclopedia/old_files/data/G044/G.0404620 Gorenstein ring
4 KB (623 words) - 19:42, 5 June 2020
$#A+1 = 119 n = 0
$#C+1 = 119 : ~/encyclopedia/old_files/data/A012/A.0102090 Alternative rings and algebras
12 KB (1,930 words) - 19:54, 15 March 2023
...prime if and only if the [[principal ideal]] generated by this element is a [[prime ideal]].
...lign="top">[2]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)</TD></TR>
1 KB (252 words) - 16:49, 13 August 2023
$#A+1 = 28 n = 0
A module $ M $
3 KB (480 words) - 16:44, 13 January 2024
$#A+1 = 20 n = 0
...s. Jacobson categories are axiomatically defined as categories $ \mathbf A $
3 KB (517 words) - 22:14, 5 June 2020
$#A+1 = 70 n = 0
Hypercomplex numbers of the form $ a + be $,
6 KB (895 words) - 19:36, 5 June 2020
A formal Dirichlet series over a ring $R$ is associated to a function $a$ from the positive integers to $R$
L(a,s) = \sum_{n=1}^\infty a(n) n^{-s}
2 KB (358 words) - 17:25, 11 November 2023
$#A+1 = 84 n = 0
...gebraic analogue of the concept of the differential of a function. Let $ A $
6 KB (834 words) - 17:32, 5 June 2020
A condition on a [[binary operation]] $\cdot$ on a set $X$: that for all $x, y \in X$
In the context of [[non-associative rings and algebras]], a flexible ring or algebra is one whose multiplication satisfies the flexible identity, which may be e
380 bytes (66 words) - 21:50, 5 January 2016
''of an algebra $A$ over a field or over a commutative associative ring $P$''
...ly. If the $d_{\xi\eta}^\zeta$ are the structure constants of the algebra $A$ in another base $\{ f_\xi : \xi \in I \}$, where $f_\xi = \sum t_\xi^\alph
1 KB (222 words) - 17:16, 31 May 2016
...converse is not true. The following properties of a finite-dimensional $K$-algebra $R$ are equivalent:
1) $R$ is a Frobenius algebra;
2 KB (316 words) - 15:37, 5 February 2016
$#A+1 = 91 n = 0
$#C+1 = 91 : ~/encyclopedia/old_files/data/Q076/Q.0706500 Quasi\AAhFrobenius ring,
7 KB (1,087 words) - 08:09, 6 June 2020
$#A+1 = 16 n = 3
By the Pontryagin–Thom theorem, there is a ring spectrum $ MU $(
4 KB (633 words) - 08:47, 26 March 2023
$#A+1 = 73 n = 0
...1 = 73 : ~/encyclopedia/old_files/data/D033/D.0303180 Discretely\AAhnormed ring,
5 KB (800 words) - 19:36, 5 June 2020
A [[module]] which is close to being [[free module|free]].
...over a ring $R$ is ''stably free'' if there exist free modules $F$ and $G$ over $R$ such that
423 bytes (70 words) - 15:03, 19 November 2023
$#A+1 = 93 n = 0
$#C+1 = 93 : ~/encyclopedia/old_files/data/L060/L.0600190 Local ring
9 KB (1,456 words) - 22:17, 5 June 2020
...nilpotent is a difficult problem; essentially only one example of such an algebra is known (see [[#References|[5]]]).
...locally nilpotent. It is not clear (1982) whether a finitely-generated nil algebra is nilpotent.
6 KB (1,020 words) - 16:48, 25 March 2023
...ng $f$ of a [[Measurable space|measurable space]] $(X_1, \mathcal A_1)$ to a measurable space $(X_2, \mathcal A_2)$ such that
f^{-1}(A) = \{x : f(x)\in A\} \in \mathcal A_1 \quad \text{ for each } A\in\mathcal A_2.
2 KB (395 words) - 04:33, 23 July 2015
$#A+1 = 42 n = 0
...1 = 42 : ~/encyclopedia/old_files/data/R081/R.0801000 Relative homological algebra
3 KB (465 words) - 08:10, 6 June 2020
...is an [[equivalence relation]]. Congruence arises when $A$, $B$ represent a [[bilinear form]] or [[quadratic form]] with respect to different bases, th
* P.M. Cohn, "Basic Algebra: Groups, Rings and Fields", Springer (2004) {{ISBN|1852335874}} {{ZBL|1003.
504 bytes (75 words) - 18:11, 14 November 2023
..., "comultiplication" $\phi : A \to A \otimes_k A$ and "counit" $\epsilon : A \to k$ such that the diagrams
...i} \\ A \otimes A & \stackrel{\phi \otimes 1}{\longrightarrow} & A \otimes A \end{array}
4 KB (657 words) - 08:10, 26 November 2023
''over a ring $R$, cyclic left module''
...f commutative rings over which every finitely generated module splits into a direct sum of cyclic modules [[#References|[2]]].
1 KB (206 words) - 17:29, 30 April 2014
$#A+1 = 66 n = 0
...eater than 1. The following normality criterion holds [[#References|[1]]]: A [[Noetherian scheme|Noetherian scheme]] $ X $
5 KB (706 words) - 15:10, 7 June 2020
$#A+1 = 52 n = 0
The quotient group of the group $ D ( A) $
5 KB (820 words) - 19:36, 5 June 2020
$#A+1 = 67 n = 1
$#C+1 = 67 : ~/encyclopedia/old_files/data/W098/W.0908060 Witt algebra
4 KB (623 words) - 08:29, 6 June 2020
$#A+1 = 55 n = 1
A [[Universal algebra|universal algebra]] which is a [[Group|group]] relative to the addition operation $ + $(
4 KB (643 words) - 08:01, 6 June 2020
...d the set $Q_{\text{l}} ( R )$ of all equivalence classes $[ A , f ]$ is a ring extension of $R$ with arithmetic defined by
\begin{equation*} [ A , f ] + [ B , g ] = [ A \bigcap B , f + g ], \end{equation*}
10 KB (1,646 words) - 19:47, 23 December 2023
====Algebra of sets====
...or [[Field of sets|field of sets]] by some authors. A collection $\mathcal{A}$ of subsets of some set $X$ which contains the empty set and is closed und
6 KB (884 words) - 09:36, 16 August 2013
An element $\zeta$ of a ring $R$ with unity $1$ such that $\zeta^m = 1$ for some $m \ge 1$. The least s
...For any $k$ that is relatively prime to $m$, the element $\zeta^k$ is also a primitive root. The number of all primitive roots of order $m$ is equal to
1 KB (238 words) - 19:08, 12 January 2016
...$\mathfrak{m}$, and in that case $( A , \mathfrak m )$ is a regular local ring.
...s of finite length; in that case $y _ { 1 } , \dots , y _ { s }$ is called a system of parameters of $M$.
3 KB (406 words) - 17:45, 1 July 2020
$#A+1 = 17 n = 0
of a group (algebra, ring, semi-group) $ X $
2 KB (240 words) - 19:37, 5 June 2020
...gebra, ring, semi-group, cf. [[Representation of a group|Representation of a group]]) $X$ in Hilbert spaces $H_1$, $H_2$, satisfying the condition
for a certain [[Unitary operator|unitary operator]] $U\colon H_1\to H_2$ and all
376 bytes (57 words) - 16:52, 16 August 2014
$#A+1 = 32 n = 7
$#C+1 = 32 : ~/encyclopedia/old_files/data/H110/H.1100300 Hopf ring
7 KB (1,053 words) - 22:11, 5 June 2020
$#A+1 = 28 n = 0
$#C+1 = 28 : ~/encyclopedia/old_files/data/S083/S.0803470 Schur ring
3 KB (500 words) - 18:57, 11 January 2024
...ural structure of a topological module over the completion $\hat R$ of the ring $R$.
...$A$ that is a $G$-module, where the multiplication mapping $G \times A \to A$ is required to be continuous.
1 KB (204 words) - 18:09, 17 April 2017
An astronomical term denoting the disposition of three celestial bodies on a line.
...zygy, between the $m_i$ is a set $(a_i)_{i\in I}$ of elements of the ring $A$ such that $\sum_{i\in I} a_i m_i = 0$. Thus there arises the module of syz
906 bytes (150 words) - 16:48, 23 November 2023
A (real or complex) number having the property that any number does not chang
$$0\cdot a = a \cdot 0 = 0 .$$
3 KB (471 words) - 21:06, 3 January 2015
...encyclopediaofmath.org/legacyimages/c/c025/c025920/c0259202.png" />
over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
...w.encyclopediaofmath.org/legacyimages/c/c025/c025920/c02592010.png" /> and
a right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath
6 KB (880 words) - 17:29, 7 February 2011
A tensor of type <img align="absmiddle" border="0" src="https://www.encyclope
...ww.encyclopediaofmath.org/legacyimages/c/c025/c025970/c02597010.png" /> be
a finite-dimensional vector space with basis <img align="absmiddle" border="0
10 KB (1,407 words) - 17:29, 7 February 2011
A transcendental extension of a field $k$ is
a [[field extension]] that is not an [[algebraic extension]]. An extension $K
3 KB (492 words) - 19:41, 13 December 2015
...images/h/h110/h110290/h11029017.png" /> can be resolved, i.e., involved in
a short [[Exact sequence|exact sequence]] of group schemes <img align="absmid
...ncyclopediaofmath.org/legacyimages/h/h110/h110290/h11029056.png" />-Galois
algebra (see [[#References|[a1]]]).
13 KB (1,801 words) - 19:17, 12 January 2018
$#A+1 = 55 n = 0
$#C+1 = 55 : ~/encyclopedia/old_files/data/M062/M.0602790 Matrix algebra,
5 KB (738 words) - 08:00, 6 June 2020
$#A+1 = 24 n = 0
A mapping establishing a correspondence between an element $ a $
3 KB (474 words) - 19:41, 5 June 2020
$#A+1 = 38 n = 0
$#C+1 = 38 : ~/encyclopedia/old_files/data/A010/A.0100860 Adjoint module,
3 KB (435 words) - 16:09, 1 April 2020
$#A+1 = 92 n = 0
...njective object]] in the category of (right) modules over an associative [[ring with identity]] $ R $,
7 KB (1,085 words) - 22:12, 5 June 2020
The commutator subgroup of a group $G$, also called the ''derived group'', second term of the [[lower ce
...ally denoted by $[G,G]$, $G'$ or $\Gamma_2(G)$. The commutator subgroup is a
3 KB (501 words) - 18:32, 12 December 2015
...y representation]] of a group or a symmetric representation of a symmetric algebra, then $\pi$ is an operator-irreducible representation if and only if $\pi$
...from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976)
1 KB (172 words) - 10:15, 13 April 2014
$#A+1 = 150 n = 0
A special type of subobject of an algebraic structure. The concept of an idea
13 KB (2,163 words) - 22:11, 5 June 2020
$#A+1 = 55 n = 0
''over a ring $ k $''
5 KB (790 words) - 22:16, 5 June 2020
...d let $F$ be the set of those closed subsets that cannot be represented as a finite union of irreducible components. If $Y\in F$, then $Y$ is reducible,
...t; for example, to the lattice of ideals in a [[Noetherian ring|Noetherian ring]].
1 KB (217 words) - 08:34, 1 August 2014
$#A+1 = 5 n = 0
...g of one of the axioms: instead of the condition $ w ( a \cdot b ) = w ( a) w ( b) $
1 KB (165 words) - 08:08, 6 June 2020
...U,A)$. Among the properties preserved by transition to a Morita-equivalent ring are the properties of being: Artinian, Noetherian, primary, simple, classic
...$S$. In particular, for $R=S$ this implies that $R$ is a [[quasi-Frobenius ring]].
3 KB (499 words) - 20:17, 26 November 2016
...are said to be invertible ideals. Each invertible ideal has a finite basis over $R$.
...="top">[1]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975)</TD></TR>
1 KB (176 words) - 18:29, 13 December 2014
$#A+1 = 133 n = 0
$#C+1 = 133 : ~/encyclopedia/old_files/data/C021/C.0201040 Cayley\ANDDickson algebra
12 KB (1,483 words) - 16:43, 4 June 2020
...cular, of natural numbers $a_1,\ldots,a_n$. The greatest common divisor of a set of numbers not all of which are zero always exists. The greatest common
...s bounded by the smaller of the degrees of the polynomials: cf [[Euclidean ring]]
4 KB (673 words) - 17:01, 26 October 2014
...tation]]), and the ring $\text{Hom}_F(P,P)$ of all linear operators on $P$ over $F$ is generated by $P$ and $G$; indeed, it is the [[cross product]] of $P$
...the dimension of $P$ over $E$ equals the dimension of $\text{Hom}_E(P,P)$ over $P$.
3 KB (470 words) - 19:10, 9 November 2016
...he maximal cardinality of a subsystem of elements which are independent in a certain sense.
...n algebraic group]] and the [[Rank of a Lie group|rank of a Lie group]] in a special way.
4 KB (631 words) - 10:02, 15 April 2014
...he theory of finite-dimensional Lie algebras should be regarded as general algebra.
...he investigation of these objects is also regarded as belonging to general algebra. The most developed theories are those of partially ordered and topological
5 KB (725 words) - 21:52, 30 March 2012
''over a field $K$''
...ak{g}$ whose elements are linear transformations of a [[vector space]] $V$ over $K$; addition of elements and multiplication of them by elements of $K$ are
3 KB (554 words) - 19:21, 24 February 2017
''of a commutative ring $A$''
...q L$ is a field extension (cf. also [[Extension of a field]]), then $L$ is a Fatou extension of $K$.
5 KB (828 words) - 11:51, 24 December 2020
$#A+1 = 89 n = 0
...ois group]]) to the case of associative rings with a unit element. Let $ A $
6 KB (975 words) - 04:16, 19 March 2022
[[Extension of a field|Extension of a field]]) based on the
divisibility of integers by a given prime number $p$. The extension is
6 KB (1,089 words) - 20:29, 9 April 2017
An [[ideal]] (of a [[ring]], algebra, [[semi-group]], or [[lattice]]) generated by one element $\alpha$, i.e. th
The left principal ideal $L(\alpha)$ of a ring $A$ contains, in addition to the element $\alpha$ itself, also all the element
3 KB (484 words) - 20:54, 28 November 2014
addition. Moreover, an associative algebra should be a vector space
over a fixed field $F$, and the multiplication in it is compatible
11 KB (1,726 words) - 20:09, 15 December 2020
The Lie algebra of an algebraic subgroup (see
a finite-dimensional vector space $V$ over a field $k$. If $\mathfrak g$ is an
3 KB (458 words) - 04:43, 4 January 2022
$#A+1 = 47 n = 0
$#C+1 = 47 : ~/encyclopedia/old_files/data/P110/P.1100170 Poisson algebra
8 KB (1,172 words) - 19:18, 17 March 2023
$#A+1 = 49 n = 0
$#C+1 = 49 : ~/encyclopedia/old_files/data/M065/M.0605490 Multiplicity of a module
4 KB (548 words) - 06:47, 16 June 2022
$#A+1 = 113 n = 0
of a ring $ A $
6 KB (990 words) - 10:59, 29 May 2020
the group of all $(n\times n)$ invertible matrices over an associative ring (cf.
with a unit; the usual symbols are $\def\GL{\textrm{GL}} \GL_n(K)$ or $\GL(n,K)$.
3 KB (592 words) - 20:32, 5 December 2014
...em]] is a [[Vector space|vector space]] $V$ over a field $K$ together with a $K$-[[trilinear mapping]] $V \times V \times V \rightarrow V$.
A vector space $V$ with triple product $[\,,\,,\,]$ is said to be a Lie triple system if
3 KB (559 words) - 16:43, 15 March 2023
...peratorname{Inn} ( R )$, the set of all $X$-inner automorphisms of $R$, is a [[Normal subgroup|normal subgroup]] of $\operatorname{Aut}( R )$.
...X$-inner. Of course, $q$ is determined by $\sigma$ up to multiplication by a non-zero element of the extended centroid $C = \mathbf{Z} ( Q ) = \mathbf{C
9 KB (1,402 words) - 13:37, 25 November 2023
...nts in a field or a commutative associative ring $K$ with a unit, which is a [[Symmetric function|symmetric function]] in its variables, that is, is inv
The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.
5 KB (801 words) - 20:34, 13 September 2016
$#A+1 = 62 n = 0
$#C+1 = 62 : ~/encyclopedia/old_files/data/O070/O.0700140 Ordered ring,
7 KB (1,019 words) - 08:04, 6 June 2020
...r algebras are the coherent algebras (cf. also [[Coherent algebra|Coherent algebra]]).
...full matrix algebra $\mathbf{C}^{n \times n}$ of $( n \times n )$-matrices over $\mathbf{C}$ such that:
7 KB (1,000 words) - 16:48, 14 January 2021
...$\phi(x)=F(x_1,\dots,x_n)$, where $F\in C[X_1,\dots,X_n]$ is a polynomial over $C$ and $x_1,\dots,x_n$ are the coordinates of an element $x\in V$ in the b
...e-dimensional vector space over a field of characteristic 0, $P(V)$ is the algebra of symmetric multilinear forms on $V$.
2 KB (276 words) - 00:18, 25 November 2018
''over a ring $A$ in commuting variables $T_1,\ldots,T_N$''
where $F_k$ is a form of degree $k$ in $T_1,\ldots,T_N$ with coefficients in $A$. The minimal value of $k$ for which $F_k \ne 0$ is called the order of the
6 KB (1,093 words) - 08:26, 16 March 2023
$#A+1 = 16 n = 0
...metrical objects on manifolds, such as vectors, tensors, forms, etc. It is a linear operator $ \nabla _ {X} $
2 KB (331 words) - 17:31, 5 June 2020
$#A+1 = 20 n = 0
''of a Banach algebra''
3 KB (528 words) - 08:22, 6 June 2020
$#A+1 = 105 n = 0
A free object in a certain class of algebraic systems.
9 KB (1,325 words) - 15:29, 17 March 2023
...ay be defined as a non-empty set $X$ with a [[binary relation]] $\ast$ and a constant $0$ satisfying the following axioms:
...e$ can then be defined by putting $x \le y$ if and only if $x \ast y = 0$. A very useful property is $(x \ast y) \ast z = (x \ast z) \ast y$.
5 KB (892 words) - 02:43, 15 February 2024
$#A+1 = 73 n = 0
$#C+1 = 73 : ~/encyclopedia/old_files/data/M062/M.0602170 Mal\AAptsev algebra,
9 KB (1,380 words) - 07:59, 6 June 2020
...t $t \in H$ is then computed in the [[free associative algebra]] (i.e. the ring of polynomials with non-commuting indeterminates $\{a_1,a_2,\ldots\}$) foll
...enerator}\\ P_{t'}P_{t''} - P_{t''}P_{t'} &\text{if}\,t=[t',t'']\,\text{is a bracketed word of weight}\ge 2.\end{cases}
3 KB (577 words) - 13:35, 20 March 2023
...he methods developed in proving it, have found applications in commutative algebra, algebraic geometry and algebraic $K$-theory; see [[#References|[a5]]] and
...Prime ideal|prime ideal]] of $R$, and denote by $Q ( R / P )$ the quotient ring of $R / P$. By the [[Nakayama lemma]] (cf. also [[Jacobson radical|Jacobson
8 KB (1,239 words) - 17:01, 1 July 2020
$#A+1 = 112 n = 0
A [[Noetherian ring|Noetherian ring]] $ R $
10 KB (1,471 words) - 14:52, 30 May 2020
$#A+1 = 109 n = 0
be a [[Normal subgroup|normal subgroup]] of a [[Finite group|finite group]] $ G $
5 KB (847 words) - 17:44, 4 June 2020
$#A+1 = 51 n = 0
The representation of a ring as the direct sum of subrings related to a given idempotent $ e $.
5 KB (706 words) - 15:56, 7 June 2020
''over a ring $R$''
...between free Lie algebras and both [[free group]]s and [[free associative algebra]]s.
3 KB (564 words) - 19:53, 15 March 2023
$#A+1 = 22 n = 0
...t is known as Ore's theorem (see [[Modular lattice|Modular lattice]]). For a group $ G $
5 KB (786 words) - 22:15, 5 June 2020
''of a module''
...riant under all endomorphisms of the ring. The socle can be represented as a direct sum of simple modules. [[Completely-reducible module]]s (semi-simple
2 KB (371 words) - 15:04, 19 November 2023
$#A+1 = 35 n = 1
of a finite-dimensional [[Vector space|vector space]] $ V $,
4 KB (558 words) - 16:18, 6 June 2020
$#A+1 = 71 n = 0
$#C+1 = 71 : ~/encyclopedia/old_files/data/S086/S.0806590 Spectrum of a ring
5 KB (778 words) - 13:13, 6 January 2024
...ng if the following basic properties are valid for all elements $a,b,c \in A$:
i) $a \oplus a = a$ (idempotent addition);
7 KB (1,026 words) - 20:42, 16 November 2023
$#A+1 = 133 n = 2
$#C+1 = 133 : ~/encyclopedia/old_files/data/R080/R.0800830 Regular ring (in the sense of von Neumann)
14 KB (2,228 words) - 08:10, 6 June 2020
$#A+1 = 95 n = 0
...ement of commutativity of multiplication, then it is possible to construct a number system from the points of $ 4 $-
11 KB (1,563 words) - 14:54, 7 June 2020
$#A+1 = 76 n = 1
A [[Module|module]] $ P $
7 KB (1,080 words) - 08:08, 6 June 2020
A type of quantization typically encountered in [[knot theory]], for example
...module]]). An $R$-module [[epimorphism]] $\phi:A \rightarrow P$ is called a Drinfel'd–Turaev quantization of $P$ if
2 KB (263 words) - 19:52, 5 March 2018
...order in $KG$ is a rank-$q$ $R$-Hopf algebra $H$ (cf. [[Hopf algebra|Hopf algebra]]) satisfying $H\otimes_R K \cong KG$ as $K$-Hopf algebras.
...alue $\xi(g)$. Then the $R$-Hopf order in $KG$ determined by $\xi$ (called a Larson order) is of the form
5 KB (919 words) - 04:20, 15 February 2024
...t contain the zero ideal. If no class of ideals is specifically mentioned, a minimal ideal is taken to be minimal in the set of all (non-zero) two-sided
...of all such elements. For example, every finite commutative semi-group is a homogroup.
6 KB (961 words) - 18:14, 7 May 2016
The set of solutions of a given system of algebraic equations. Let $k$
be a field and let $\bar k$ be its algebraic closure. A subset $X$ of the
4 KB (616 words) - 21:49, 30 March 2012
$#A+1 = 44 n = 1
$#C+1 = 44 : ~/encyclopedia/old_files/data/R082/R.0802430 Ring with divided powers
4 KB (663 words) - 08:28, 20 January 2024
...ian valuation. See also [[Henselization of a valued field|Henselization of a valued field]]; [[Ramification theory of valued fields|Ramification theory
...s expressing the validity of the [[Hensel lemma|Hensel lemma]]). Moreover, a complete field need not be Henselian if its value group is not Archimedean.
3 KB (405 words) - 19:33, 28 April 2014
$#A+1 = 26 n = 0
...haracter of a finite\AAhdimensional representation of a semi\AAhsimple Lie algebra
2 KB (326 words) - 07:50, 21 January 2022
...representations. If $A$ is an algebra with a unit element or a semi-simple algebra, then its regular representations are faithful (cf. [[Faithful representati
...lar representation of a group $G$ is a linear representation $R$ of $G$ on a space $E$ of complex-valued functions on $G$, defined by the formula
2 KB (386 words) - 07:53, 30 November 2014
...$\mathbf{R} ^ { m }$ correspond to the choice $A = \mathbf{R}$, while the algebra $\mathcal{D} = \mathbf{R}\cdot 1 \oplus e \cdot \mathbf{R}$, $e ^ { 2 } = 0
...A real unital finite-dimensional commutative algebra $A$ is called a Weil algebra if it is of the form
7 KB (1,051 words) - 06:27, 15 February 2024
...y inclusion contains a maximal element. Submodules and quotient modules of a Noetherian module are Noetherian. If, in an [[exact sequence]]
...ing]] is Noetherian if and only if it is finitely generated. A module has a [[Composition sequence|composition series]] if and only if it is both [[Art
919 bytes (137 words) - 19:05, 25 October 2014
The Dirac algebra arises from Dirac's solution [[#References|[a3]]] to the relativistic elect
...any unitary matrix $S$ and post-multiplying by $S ^ { - 1 }$ will produce a new set of matrices satisfying the conditions. The defining relations are o
7 KB (926 words) - 22:43, 12 December 2020
''of a free associative algebra''
...F)$ for some natural number $n$. Every $T$-ideal $T$ of a free associative algebra is primary.
4 KB (732 words) - 08:33, 29 August 2014
$#A+1 = 71 n = 0
be an associative ring with unit, and $ M $
4 KB (674 words) - 07:14, 12 July 2022
The ''Clifford algebra'' of a [[quadratic form]] is
...r of the pair $(E,Q)$) one means the quotient algebra $C(Q)$ of the tensor algebra $T(E)$ of the $K$-module $E$ by the two-sided ideal generated by the elemen
5 KB (920 words) - 17:17, 9 October 2016
...This is equivalent to the existence over $R$ of two rectangular matrices $A$ and $B$ for which
...ism into a skew-field (in particular, over commutative rings), the rank of a free module is always uniquely defined.
2 KB (351 words) - 12:24, 14 February 2020
$#A+1 = 71 n = 0
$#C+1 = 71 : ~/encyclopedia/old_files/data/E037/E.0307080 Exterior algebra,
7 KB (1,013 words) - 19:38, 5 June 2020
$#A+1 = 91 n = 0
A bijective mapping $ \kappa $
6 KB (878 words) - 14:59, 2 May 2023
$#A+1 = 13 n = 0
A set $ M $
2 KB (242 words) - 16:40, 31 March 2020
$#A+1 = 49 n = 0
x ^ {n} \equiv a ( \mathop{\rm mod} m ) ,
5 KB (717 words) - 08:27, 6 June 2020
or their subgroups, given certain special extensions of the ground ring $ R $(
...(cf. [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]) and let $ R[t _{1} \dots t _{n} ] $
7 KB (972 words) - 19:10, 5 April 2023
Two subextensions $A$ and $B$ of an extension $\def\O{\Omega}\O$ of $k$ are called linearly disj
subalgebra generated by $A$ and $B$ in $\O$ is (isomorphic to) the
1 KB (259 words) - 22:07, 5 March 2012
$#A+1 = 129 n = 0
$#C+1 = 129 : ~/encyclopedia/old_files/data/W098/W.0908080 Witt ring
9 KB (1,359 words) - 08:29, 6 June 2020
...hat discrete versions of the [[Bellman equation]] can be treated as linear over idempotent semi-rings. The so-called (finite-dimensional) generalized stati
...interesting numerical algorithm, then there is a good chance that its semi-ring analogues are important and interesting as well" , [[#References|[a9]]], [[
7 KB (1,079 words) - 20:39, 16 November 2023
$#A+1 = 68 n = 0
A submodule $ A $
4 KB (701 words) - 08:04, 21 January 2024
''on a product of modules $V\times W$''
A
6 KB (1,157 words) - 08:58, 9 December 2016
$#A+1 = 235 n = 0
''logarithmic norm, norm on a field''
14 KB (2,135 words) - 08:27, 6 June 2020
on a left $R$-module $X$'' is
a mapping $\def\phi{\varphi}\phi:X\times X \to R$ that is linear in the first
5 KB (831 words) - 17:13, 9 October 2016
where $x,y,\dots,w$ are variables and $A,B,\dots,D$ (the ''coefficients'' of the polynomial) and $k,l,\dots,t$ (the
...zero powers. When the polynomial has one, two or three terms it is called a monomial, binomial or trinomial.
9 KB (1,497 words) - 10:44, 27 June 2015
...n be identified in a natural way with the ideals (cf. [[Ideal|Ideal]]) of $A$.
...ubstitute for unique factorization into prime factors if factorization in $A$ is not unique.
5 KB (853 words) - 12:16, 22 August 2014
$#A+1 = 137 n = 0
An algebra over a field for which certain polynomial identities are true.
15 KB (2,252 words) - 08:04, 6 June 2020
...the general case the local uniformization theorem implies the existence of a finite resolving system (see [[#References|[3]]]).
...align="top">[4]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001
3 KB (440 words) - 14:25, 1 May 2014
$#A+1 = 157 n = 0
...als, that is, in Artinian rings and algebras (cf. [[Artinian ring|Artinian ring]]), and that the description of Artinian semi-simple rings and algebras coi
16 KB (2,540 words) - 08:09, 6 June 2020
$#A+1 = 44 n = 0
$#C+1 = 44 : ~/encyclopedia/old_files/data/D033/D.0303680 Division algebra
5 KB (706 words) - 09:14, 28 June 2022
$#A+1 = 85 n = 4
$#C+1 = 85 : ~/encyclopedia/old_files/data/T092/T.0902240 Tate algebra
8 KB (1,161 words) - 08:25, 6 June 2020
...algebras]]; [[Jordan algebra|Jordan algebra]]; [[Mal'tsev algebra|Mal'tsev algebra]]).
...ive (Lie, special Jordan) algebra over a field can be imbedded in a simple algebra of the same type. In some classes of algebras there are many simple algebra
16 KB (2,433 words) - 21:48, 5 January 2016
An element $e_d$ of the group ring of the [[Symmetric group|symmetric group]] $S_m$ defined by the [[Young tab
...metrizer is that it is proportional to a primitive idempotent of the group algebra $\mathbf CS_m$. The coefficient of proportionality is equal to the product
961 bytes (180 words) - 13:01, 10 August 2014
...and $B$ be finite-dimensional algebras over a field $k$, and assume that $A$ is simple and $B$ is central simple (cf. also
[[Simple algebra|Simple algebra]];
4 KB (552 words) - 16:03, 26 April 2012
$#A+1 = 100 n = 0
$#C+1 = 100 : ~/encyclopedia/old_files/data/B110/B.1100640 Blow\AAhup algebra
7 KB (941 words) - 10:59, 29 May 2020
$#A+1 = 49 n = 0
$#C+1 = 49 : ~/encyclopedia/old_files/data/A010/A.0100710 Additive theory of ideals
7 KB (1,035 words) - 20:23, 4 April 2020
$#A+1 = 59 n = 0
The theorem stating that for any way of reducing a quadratic form (cf. also [[Quadratic forms, reduction of|Quadratic forms, r
5 KB (659 words) - 06:06, 23 April 2024
$#A+1 = 55 n = 0
A [[Bilinear form|bilinear form]] $ f $
4 KB (640 words) - 03:29, 21 March 2022
...[Representation of an associative algebra|Representation of an associative algebra]]).
...ated as a module over its centre $Z(R)$, which is a commutative [[Artinian ring]].
7 KB (1,122 words) - 18:14, 3 April 2024
$#A+1 = 47 n = 0
A finitely-additive [[Measure|measure]] with real or complex values on some s
3 KB (492 words) - 04:40, 9 May 2022
$#A+1 = 35 n = 0
...cyclopedia/old_files/data/D032/D.0302310 Differentiation along the flow of a dynamical system
3 KB (469 words) - 19:35, 5 June 2020
...g, such that for any additive mapping $\mathrm{Ob}(C) \to G$, there exists a unique homomorphism $\xi: K(C) \to G$ that satisfies the condition $\phi=\x
...d can be given by generators — to each object $L \in C$, there corresponds a generator $[L]$ — and by the relations $[L]-[N]-[M]=0$ for each exact seq
4 KB (701 words) - 06:11, 26 March 2023
A ''field'' is a commutative, associative ring containing a unit in which the set of
non-zero elements is not empty and forms a group under multiplication
6 KB (929 words) - 00:28, 18 May 2013
''of a [[group]] $G$''
...(G)$, where $Z(G)$ is the centre of $G$ (cf. [[Centre of a group|Centre of a group]]). Automorphisms that are not inner are called ''outer automorphisms
2 KB (338 words) - 21:00, 29 November 2014
$#A+1 = 60 n = 0
A function given by an algebraic equation. Let $ F ( X _ {1} \dots X _ {n}
5 KB (768 words) - 22:11, 5 June 2020
$#A+1 = 75 n = 0
$#C+1 = 75 : ~/encyclopedia/old_files/data/U095/U.0905060 Unary algebra,
6 KB (889 words) - 08:27, 6 June 2020
...ible set, cf. also [[Completely-reducible set|Completely-reducible set]]). A completely-irreducible representation is (topologically) irreducible and op
1 KB (149 words) - 10:13, 13 April 2014
$#A+1 = 77 n = 0
linear form, on a unitary $ A $-
6 KB (855 words) - 20:46, 16 January 2024
$#A+1 = 35 n = 0
A subset $ T $
5 KB (722 words) - 18:41, 5 October 2023
An outer measure is a [[Set function|set function]] $\mu$ such that
...o called $\sigma$-ideal) of subsets of a given space $X$, i.e. a $\sigma$-ring $\mathcal{R}\subset \mathcal{P} (X)$ with the property that for every $E\i
9 KB (1,397 words) - 10:08, 16 August 2013
$#A+1 = 5 n = 0
...pological vector space is called indecomposable if it is not equivalent to a topological direct sum of non-zero representations of the same algebraic sy
3 KB (399 words) - 22:12, 5 June 2020
$#A+1 = 224 n = 2
A [[Linear representation|linear representation]] of the group $ S _ {m} $
18 KB (2,500 words) - 13:04, 18 February 2022
$#A+1 = 23 n = 0
over the integral group ring $ \mathbf Z [ H ] $(
2 KB (302 words) - 08:01, 6 June 2020
...ferences|[a5]]], B. Eckmann and A. Schöpf defined an injective envelope of a module to be an embedding $M \subset E$ with $E$ injective and with $M$ ess
...ope. If $R$ is a [[Ring|ring]], Bass proved that every left $R$-module has a projective cover if and only if every flat left $R$-module is projective (c
6 KB (1,031 words) - 16:55, 1 July 2020
...a Lie algebra $ \mathfrak{g} $ over a commutative ring $ \mathbb{k} $ with a unit element''
...mathbb{k} $-algebra $ U(\mathfrak{g}) $ with a unit element, together with a mapping $ \sigma: \mathfrak{g} \to U(\mathfrak{g}) $ for which the followin
6 KB (970 words) - 18:59, 5 April 2023
$#A+1 = 48 n = 0
over a commutative ring $ K $.
5 KB (636 words) - 22:11, 5 June 2020
$#A+1 = 27 n = 0
be a division ring (cf. [[Division algebra|Division algebra]]) with centre $ k $
2 KB (328 words) - 13:15, 7 April 2023
$#A+1 = 57 n = 0
A set $ K $
5 KB (680 words) - 22:17, 5 June 2020
$#A+1 = 25 n = 0
A module $ A $
2 KB (330 words) - 16:25, 23 November 2023
$#A+1 = 47 n = 0
$#C+1 = 47 : ~/encyclopedia/old_files/data/L059/L.0509040 Linear algebra
9 KB (1,394 words) - 08:15, 9 January 2024
...he $\sigma$-algebra generated by the open sets (i.e. the smallest $\sigma$-algebra of
...ompact sets, see {{Cite|Hal}}. Under suitable assumptions, for instance on a separable locally compact metric space, the two notions coincide.
5 KB (795 words) - 09:26, 7 December 2012
$#A+1 = 92 n = 0
...to addition (in particular, the ring has a zero element, denoted by 0, and a negative element $ - x $
21 KB (3,225 words) - 09:25, 13 July 2022
''of degree (order) $\def\SL{\textrm{SL}}\def\GL{\textrm{GL}} n$ over a ring $R$''
...tance here: the ordinary determinant in the case when $R$ is a commutative ring, the non-commutative Dieudonné determinant (cf.
6 KB (988 words) - 22:16, 10 January 2015
...called a [[formally real field]]. The field of real numbers is a model of a formally real field. More generally, every ordered field is formally real.
...al of odd degree over $k$ has a root in $k$. Every formally-real field has a real-closed ordered algebraic extension.
2 KB (375 words) - 19:58, 7 December 2014
$#A+1 = 139 n = 0
...cified class of objects in this category. The categories of modules over a ring form the principal range of application of this concept.
11 KB (1,715 words) - 22:10, 5 June 2020
A
[[Universal algebra|universal algebra]] that is a topological space such that the algebraic operations are continuous.
9 KB (1,142 words) - 17:15, 15 November 2023
$#A+1 = 17 n = 0
...= 17 : ~/encyclopedia/old_files/data/I052/I.0502330 Invariant subspace of a representation
1 KB (174 words) - 22:13, 5 June 2020
$#A+1 = 25 n = 0
...1 = 25 : ~/encyclopedia/old_files/data/P075/P.0705300 Projective object of a category
4 KB (597 words) - 08:08, 6 June 2020
...ots,x_n)$ a cubic form with coefficients in $K$ (one calls it a cubic form over $K$). The equation
...e $\mathbf{P}^n(K)$, so that the algebraic-geometric theory of cubic forms over an [[algebraically closed field]] $K$ is reduced to the theory of cubic hyp
3 KB (526 words) - 11:44, 12 October 2023
''of a Lie group''
...cf. also [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]).
8 KB (1,184 words) - 16:59, 1 July 2020
A field with a finite number of elements. First considered by E. Galois [[#References|[1]]
...extensions of any Galois field within its fixed algebraic closure is such a lattice.
4 KB (749 words) - 18:32, 2 November 2014
A pre-$\lambda$-ring is a commutative ring $R$ with identity element $1$ and a set of mappings $\lambda^n : R \rightarrow R$, $n = 0,1,2,\ldots$ such that
...representation ring $R(G)$ of a finite group $G$ (cf. [[Representation of a compact group]]). In all these cases the $\lambda^n$ are induced by taking
10 KB (1,721 words) - 07:44, 23 March 2016
$#A+1 = 73 n = 0
...aic structure. Maximal ideals play an essential role in ring theory. Every ring with identity has maximal left (also right and two-sided) ideals. The quoti
7 KB (1,160 words) - 08:00, 6 June 2020
$#A+1 = 17 n = 0
$#C+1 = 17 : ~/encyclopedia/old_files/data/C023/C.0203370 Commutative algebra
16 KB (2,400 words) - 17:45, 4 June 2020
$#A+1 = 73 n = 0
$#C+1 = 73 : ~/encyclopedia/old_files/data/U095/U.0905630 Universal algebra
11 KB (1,662 words) - 04:57, 24 February 2022
$#A+1 = 91 n = 0
''of a vector space''
8 KB (1,152 words) - 08:29, 6 June 2020
''restricted Lie algebra''
An algebra $ L $
10 KB (1,443 words) - 10:14, 17 December 2019
$#A+1 = 41 n = 0
A [[Complete lattice|complete lattice]] $ L = \langle L , \lor , \wedge \r
8 KB (1,218 words) - 08:02, 6 June 2020
$#A+1 = 24 n = 1
be a free complex of left $ A $-
3 KB (446 words) - 08:10, 6 June 2020
A ''sesquilinear form'' is a function in two variables on a
...n automorphism $\def\s{\sigma}a\mapsto a^\s$, is a mapping $q:E\times E\to A$, $(x,y)\mapsto q(x,y)$, linear in $x$ for fixed $y$, and semi-linear in $y
3 KB (572 words) - 13:14, 7 April 2023
...functors on various categories of algebraic objects (modules over a given ring, sheaves, etc.).
...nalytic geometry (with the difference that the transition from geometry to algebra in homology theory is irreversible).
12 KB (1,885 words) - 23:48, 23 April 2017
...a square matrix $A = (a_{ij})$ of order $n$ over a commutative associative ring $R$ with unit 1'' is
where $i_1,\dots,i_n$ is a permutation of the numbers $1,\dots,n$ and $k$ is the number of inversions
11 KB (1,876 words) - 20:27, 30 November 2016
$#A+1 = 61 n = 0
A mapping $ f $
4 KB (590 words) - 07:45, 7 January 2024
$#A+1 = 45 n = 0
''of a matrix $ F ( x) $
3 KB (520 words) - 17:30, 16 December 2020
$#A+1 = 131 n = 0
...]. Originally (1842), Ch. von Staudt defined, on the projective line/plane over $ \mathbf C $,
9 KB (1,440 words) - 16:43, 4 June 2020
...ectrum of a ring|Spectrum of a ring]]) and $X=P_k^n$ is a projective space over $k$, then the set of rational $k$-points of $\operatorname{Hilb}(P_k^n/k)$
...{Hilb}(X/S)$ is the direct sum of the schemes $\operatorname{Hilb}^P(X/S)$ over all $P\in\mathbf Q(z)$. For any connected ground scheme $S$ the scheme $\op
3 KB (484 words) - 05:48, 17 April 2024
$#A+1 = 39 n = 0
'' $ A $
3 KB (505 words) - 08:09, 6 June 2020
...ntly, is invariant under the transformation $\alpha \mapsto \alpha^{-1}$). A Lie group $G$ is unimodular if and only if
...athfrak{g}$), where $\mathrm{ad}$ is the adjoint representation of the Lie algebra $\mathfrak{g}$ of $G$. Any compact, discrete or Abelian locally compact gro
2 KB (238 words) - 18:19, 12 October 2014
An algebraic analogue of the concept of a local Lie group (cf. [[Lie group, local|Lie group, local]]). The theory of
A formal group over a field $ k $
17 KB (2,537 words) - 22:38, 15 December 2019
A ''scheme'' is a
...nsists of a topological space $X$ (the underlying space of the scheme) and a
12 KB (2,191 words) - 13:20, 24 November 2013
$#A+1 = 119 n = 2
$#C+1 = 119 : ~/encyclopedia/old_files/data/B016/B.0106920 Boolean algebra,
14 KB (2,077 words) - 21:17, 17 January 2021
The same rule applies also to several other situations. The following is a list of rather important cases.
...he maps $f,g U \to \mathbb R$ are differentiable along a [[Vector field on a manifold|vector field]]. The rule becomes then
5 KB (757 words) - 10:34, 11 December 2013
$\newcommand{\frakA}{\mathfrak{A}}$
...]] with the distributive action of a ring. A module is a generalization of a (linear)
23 KB (3,918 words) - 04:31, 23 July 2018
A '''linear algebraic equation''' (or as well a '''linear equation''') is an
unknowns and need to be found. A solution of the system of linear
12 KB (2,085 words) - 22:05, 5 March 2012
$#A+1 = 250 n = 0
A special case of cohomology of algebras. Let $ \mathfrak G $
21 KB (3,027 words) - 17:45, 4 June 2020
...s the [[Localization in a commutative algebra|localization]] of the ring $ A $ with respect to the multiplicative system $ \{ f^{n} \}_{n \in \Bbb{N}_{0
...ieck ([[#References|[1]]]), who created the theory of schemes. A scheme is a ringed space that is locally isomorphic to an affine scheme.
6 KB (884 words) - 09:29, 13 December 2016
A ''Schubert variety''
.... In Grassmann coordinates these conditions are given by linear equations; a Schubert variety is an irreducible (generally speaking, singular) algebraic
3 KB (391 words) - 16:26, 9 December 2023
...be a finitely-generated $A$-module with $\dim_AM = s$. Then $M$ is called a Buchsbaum module if the difference
\begin{equation*} \operatorname{l} _ { A } ( M / \mathfrak{q}M ) - e _ { \mathfrak{q} } ^ { 0 } ( M ) \end{equation*
27 KB (4,003 words) - 17:43, 1 July 2020
$#A+1 = 14 n = 0
finitely generated over some field $ k $)
2 KB (246 words) - 19:37, 5 June 2020
...set of basic commutators constructed in this way is a base of the free Lie algebra with $R$ as set of free generators [[#References|[1]]].
...TD> <TD valign="top"> A.I. Shirshov, "On bases of free Lie algebras" ''Algebra i Logika'' , '''1''' : 1 (1962) pp. 14–19 (In Russian)</TD></TR></tab
3 KB (539 words) - 08:07, 13 February 2024
$#A+1 = 182 n = 0
is a polynomial in $ x $
13 KB (2,001 words) - 02:12, 1 March 2022
A ''semi-linear mapping'' is a mapping $\def\a{\alpha}\a$ from a (left)
[[Module|module]] $M$ into a (left) module $N$ over the same ring $A$, satisfying the conditions
2 KB (264 words) - 22:23, 2 November 2013
A branch of algebra, dealing mainly with the study of the so-called $K$-functors ($K_0, K_1$, e
...ra appear in topology. A vector space can be regarded as a special case of a
14 KB (2,405 words) - 22:14, 10 January 2015
...ld]], and let $F(M)$ be the algebra of smooth real-valued functions on it. A tangent vector to $M$ at $m\in M$ is an $\mathbb{R}$-linear mapping $v:F(M)
For this definition one can equally well (in fact, better) use the ring of germs of smooth functions $F(M,m)$ on $M$ at $m$.
8 KB (1,509 words) - 05:24, 22 May 2017
...he following general principle: The functor $ X \mapsto \{ \textrm{ the algebra of functions on } X \} $
...nslate the definition of a group into the language of algebras: instead of a space $ G $
18 KB (2,674 words) - 19:09, 16 December 2019
$#A+1 = 40 n = 5
$#C+1 = 40 : ~/encyclopedia/old_files/data/S086/S.0806210 Space over an algebra
8 KB (1,172 words) - 16:55, 7 June 2020
...y subgroup of finite index in $\def\O{\mathcal{O}}G_\O$, where $\O$ is the ring of integers in an algebraic number field $k$ (cf.
[[Algebraic number theory|Algebraic number theory]]) and $G$ is a connected
4 KB (613 words) - 21:48, 5 March 2012
$#A+1 = 101 n = 0
$#C+1 = 101 : ~/encyclopedia/old_files/data/N067/N.0607360 Norm on a field
6 KB (1,003 words) - 21:35, 13 January 2021
''bi-algebra, hyperalgebra''
A graded module $ A $
11 KB (1,576 words) - 08:51, 16 December 2019
...utative-associative ring $A$ with a unit, then $\Pf X$ is the element of $A$ given by the formula
where the summation is over all possible partitions $s$ of the set $\{1,\ldots,2n\}$ into non-inte
1 KB (251 words) - 19:57, 30 November 2014
A field extension $K$ is a field containing a given field $k$ as a subfield. The notation $K/k$
Let $K/k$ and $L/k$ be two extensions of a field $k$. An isomorphism of
6 KB (1,030 words) - 17:29, 12 November 2023
...ces|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $ A $
...e, commutative ring with unit element. Witt vectors with components in $ A $
17 KB (2,502 words) - 17:25, 22 December 2019
...ng $A$ with a unit is the $A$-module $V_1 \tensor_A V_2$ together with an $A$-bilinear mapping
...times V_2 \to W$, where $W$ is an arbitrary $A$-module, there is a unique $A$-linear mapping $b : V_1 \tensor_A V_2 \to W$ such that
11 KB (1,992 words) - 03:52, 23 July 2018
====Discriminant of a [[polynomial]]====
...f(x)=a_0 x^n+a_1 x^{n-1}+\cdots + a_n$, $a_0 \ne 0$, whose roots are $\def\a{ {\alpha}}\a_1,\dots,\a_n$ is the product
16 KB (2,947 words) - 08:53, 9 December 2016
$\newcommand{\A}{\mathcal A}
...formally, it is a pair $(X,\A)$ consisting of a set $X$ and a σ-algebra $\A$ of subsets of $X$.
15 KB (2,605 words) - 07:10, 23 September 2012
..._{n} $ on $ \mathbf{P}_{\mathbb{k}}^{n} $, a projective scheme is given by a system of homogeneous algebraic equations:
...is complete (compact in the case $ \mathbb{k} = \mathbb{C} $). Conversely, a complete scheme is projective if there is an [[Ample sheaf|ample]], [[Inver
3 KB (412 words) - 13:07, 17 April 2023
$#A+1 = 209 n = 2
be a set of generators for a finitely-generated group $ G $.
19 KB (2,908 words) - 20:20, 12 January 2024
$#A+1 = 66 n = 0
A vector subbundle $ \pi : P \rightarrow M $
5 KB (750 words) - 08:06, 6 June 2020
...which asks which Banach algebras have a unique complete algebra norm. For a substantial recent (as of 2000) account of automatic continuity theory for
...raised in [[#References|[a12]]]. A positive result is known in many cases; a striking sufficient condition, involving analytic functions of several comp
11 KB (1,760 words) - 16:56, 1 July 2020
A
...nected by the relation $\norm{x^* x} = \norm{x}^2$ for any element $x \in A$. $\Cstar\!$-algebras were introduced in 1943
15 KB (2,316 words) - 21:05, 20 April 2012
...ules over $\Gamma$ is an [[Abelian category]], so one can do [[homological algebra]] with these objects.
...imilarly, equivariant local cohomology can be described using modules over a category depending on the space in question.
3 KB (473 words) - 18:52, 28 October 2016
''dual quasi-triangular Hopf algebra, co-quasi-triangular Hopf algebra, quantum group''
...tain axioms such that the [[Category|category]] of (co-) modules of $H$ is a [[Braided category|braided category]] (cf. also [[Quantum groups|Quantum gr
14 KB (2,048 words) - 18:24, 3 August 2020
...ith exact infinite products, and $M : \mathcal{C} \rightarrow \mathcal{A}$ a covariant [[Functor|functor]]. Define the objects $C ^ { n } ( \mathcal{C}
where $( \alpha _ { 1 } , \dots , \alpha _ { n } )$ is a sequence of morphisms of $\mathcal{C}$ with $\operatorname { codom} \alpha
9 KB (1,283 words) - 20:55, 8 February 2024
...6366011.png" />, instead of an isomorphism or
a homomorphism one speaks of
a (metric) automorphism or an endomorphism.
...ww.encyclopediaofmath.org/legacyimages/m/m063/m063660/m06366015.png" /> be
a metric isomorphism; then it is said that <img align="absmiddle" border="0"
8 KB (1,175 words) - 05:58, 15 May 2012
...; [[Vector bundle|Vector bundle]]; [[Tangent bundle|Tangent bundle]]). For a comprehensive treatment and lists of references, see [[#References|[a8]]],
...ooth global sections of $A$, and a morphism of vector bundles $q _ { A } : A \rightarrow T M$, where $T M$ is the tangent bundle of $M$, called the anch
12 KB (1,916 words) - 16:55, 1 July 2020
''Jordan–Banach algebra''
...a|Jordan algebra]] over the field of real or complex numbers, endowed with a complete norm $|.|$ satisfying
13 KB (1,881 words) - 16:46, 1 July 2020
$#A+1 = 59 n = 0
...ation of their arity. An algebraic system (a [[Universal algebra|universal algebra]]) with signature $ \Omega $
4 KB (651 words) - 18:47, 13 January 2024
$#A+1 = 172 n = 1
$#C+1 = 172 : ~/encyclopedia/old_files/data/A011/A.0101300 Alexander invariants
12 KB (1,694 words) - 06:42, 26 March 2023
$#A+1 = 163 n = 0
...mum), that is, on the concept of an [[Idempotent semi-ring|idempotent semi-ring]]. In particular, idempotent analysis deals with functions taking values in
18 KB (2,598 words) - 22:11, 5 June 2020
...Subsequently, Hilbert's basis theorem was extensively used in commutative algebra.
be an irreducible polynomial over the field $ \mathbf Q $
18 KB (2,720 words) - 19:17, 19 December 2019
$#A+1 = 60 n = 4
A free $ \mathbf Z _ {p} $-
4 KB (651 words) - 08:25, 6 June 2020
...[tilting module]] over a finite-dimensional hereditary algebra (cf. also [[Algebra]]; [[Endomorphism]]).
...hout oriented cycles. A finite-dimensional $H$-module ${}_{H} T$ is called a tilting module if
8 KB (1,215 words) - 19:50, 24 December 2023
$#A+1 = 109 n = 0
of a topological space $ X $
9 KB (1,321 words) - 17:45, 4 June 2020
$#A+1 = 1 n = 0
$#C+1 = 1 : ~/encyclopedia/old_files/data/A011/A.0101340 Algebra
17 KB (2,478 words) - 16:09, 1 April 2020
$#A+1 = 138 n = 1
''monad, on a category $ \mathfrak R $''
14 KB (2,048 words) - 09:29, 3 July 2021
$#A+1 = 76 n = 1
A tensor field $ A $(
8 KB (1,160 words) - 08:05, 6 June 2020
$#A+1 = 43 n = 0
are two algebraically-irreducible representations of some group or algebra in two vector spaces $ X $
5 KB (720 words) - 08:12, 6 June 2020
$#A+1 = 30 n = 0
A mapping of an algebraic system $ A $
2 KB (350 words) - 19:37, 5 June 2020
...ly one fixes a quotient module $X/A$, that is, an extension of the module $A$ by the module $B$ is an [[exact sequence]]
0 \rightarrow A \rightarrow X \rightarrow B \rightarrow 0 \ .
2 KB (354 words) - 21:55, 30 October 2016
$#A+1 = 63 n = 0
A function $ f $
8 KB (1,125 words) - 07:04, 29 March 2024
...e or more equal parts of a unit. It is denoted by the symbol $a/b$, where $a$ and $b\ne 0$ are integers (cf.
...$b$. A fraction may also be considered as the ratio produced by dividing $a$ by $b$.
4 KB (649 words) - 17:15, 9 December 2013
[[Category:Linear and multilinear algebra; matrix theory]]
A matrix is a rectangular array
18 KB (3,377 words) - 17:54, 2 November 2013
$#A+1 = 39 n = 0
...is $ \{ e _ {1}, \dots, e _ {n} \} $ of a module $ E $ with respect to a form $ f $''
2 KB (305 words) - 19:52, 1 February 2022
An [[Algebra|algebra]] in which the identities $$
...m mechanics (cf. also [[#References|[2]]]), and later found application in algebra, analysis and geometry.
13 KB (1,800 words) - 19:51, 15 March 2023
...} ( X )$ vanishing on $E$ can be approximated by functions $g \in \mathcal{A} ( X )$ vanishing near $E$.
...00) whether (in specific cases) each set of spectral synthesis is actually a Ditkin set (this problem may be called the synthesis-Ditkin problem; in [[#
10 KB (1,614 words) - 16:56, 1 July 2020
$#A+1 = 59 n = 0
...ints of which are normal, that is, are integrally-closed integral domains. A point $ x $
8 KB (1,143 words) - 16:48, 19 February 2022
$#A+1 = 230 n = 1
H ^ {n} ( R, A) = \
19 KB (2,870 words) - 09:48, 26 March 2023
A [[binary operation]] on compatible [[matrix|matrices]] over a [[ring]] $R$. There are several such operations.
...red to as "matrix multiplication". The product of an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ is the $m \times p$ matrix $AB$ with entrie
2 KB (289 words) - 19:22, 11 November 2023
...inite is essential, but it may be omitted if the variety contains a finite algebra with more than one element.
...sal algebras $M$ of signature $\Omega$ commute if and only if there exists a ternary [[Term|term]] $f$ of the signature $\Omega$ such that
10 KB (1,593 words) - 01:28, 17 January 2017
A branch of algebra which deals with objects which involve not only addition and multiplication
One of the main objects of differential algebra is the algebra of differential polynomials $ {\mathcal F} \{ Y _{1} \dots Y _{n} \} $ ,
30 KB (4,468 words) - 18:44, 17 December 2019
...[[#References|[a1]]] to certain convergent formal power series rings over a field of non-zero characteristic.
The following assertion holds: A Noetherian local ring has AP if and only if it is excellent and Henselian.
16 KB (2,447 words) - 00:27, 15 February 2024
$#A+1 = 83 n = 1
A complex-valued function $ \phi $
7 KB (1,019 words) - 20:29, 16 January 2024
...bilinear form $\Phi$ on a left $K$-module $E$, where $K$ is a commutative ring (cf.
...x of any automorphism in $\Sp_{2m}(K)$ with respect to $\{e_i\}$ is called a symplectic matrix.
6 KB (1,078 words) - 14:22, 3 November 2013
A formula for computing
...$ u $ and $ v $, and $ \widehat{L} $ is the closure of $ L $ in $ \widehat{A} $. Then the mapping
6 KB (1,020 words) - 17:41, 4 May 2017
Here $G_k$ is a [[Linear algebraic group|linear algebraic
group]], defined over a [[Global field|global field]] $k$, $V$ is the
6 KB (1,071 words) - 06:53, 28 December 2021
$#A+1 = 92 n = 2
be a fixed natural number and $ K _ {m} $
7 KB (1,035 words) - 05:58, 19 March 2022
is a formal calculus of symbols representing geometric conditions used to solve
Justifying Schubert's enumerative calculus was a major theme of twentieth century
8 KB (1,263 words) - 08:49, 30 March 2012
$#A+1 = 232 n = 9
$#C+1 = 232 : ~/encyclopedia/old_files/data/W097/W.0907670 Weyl algebra
28 KB (4,182 words) - 19:30, 19 January 2024
$#A+1 = 25 n = 0
A subset of points of a projective space $ P ^ {n} $
3 KB (364 words) - 08:09, 13 July 2022
$#A+1 = 39 n = 1
...o non-trivial algebraic extension (cf. [[Extension of a field|Extension of a field]]) can be ordered. E. Artin and O. Schreier ([[#References|[a1]]]) ch
8 KB (1,240 words) - 08:10, 6 June 2020
...id $R$ if for any $\lambda \in R$ and $a \in A$ the product $\lambda a \in A$ is defined, such that
\lambda (\mu a) = (\lambda\mu) a
6 KB (1,055 words) - 05:59, 22 April 2023
...s of the second class are used most often in [[Universal algebra|universal algebra]], and in automatic theorem proving procedures.
...number of argument places of $f$. If $\rho ( f ) = 0$, then $f$ is called a constant.
9 KB (1,433 words) - 17:00, 1 July 2020
...alpha)$, where $\alpha$ is the root of a polynomial of the form $X^p - X - a$, an Artin–Schreier polynomial.
...ne since $A(x) = A(x+1)$. It is in fact $\mathbf{F}_p$-linear on $F$ as a [[vector space]], with kernel the one-dimensional subspace generated by $
2 KB (380 words) - 19:41, 15 November 2023
$#A+1 = 31 n = 0
and of a rank lower than $ t $,
4 KB (563 words) - 17:28, 26 December 2020
...cycles in the co-invariant algebra, which is isomorphic to the cohomology ring of $\mathcal{F}_n$ [[#References|[a6]]]:
...ubert polynomials, and [[#References|[a13]]] and [[#References|[a20]]] for a more geometric treatment.
10 KB (1,494 words) - 09:23, 10 November 2023
...spaces|Spectrum of spaces]]). They immediately found wide applicability in a variety of areas of [[Homotopy|homotopy]] theory, most notably in the stabl
...atural right action of the [[Steenrod algebra|Steenrod algebra]] $\mathcal{A}$ which is unstable: at the prime $2$, for example, this means that
7 KB (996 words) - 11:15, 20 January 2021
''of a field $k$''
...p to isomorphism, exactly one division algebra over $k$ (i.e. a skew-field over $k$).
7 KB (1,232 words) - 12:12, 30 December 2015
$#A+1 = 218 n = 1
...ncyclopedia/old_files/data/R081/R.0801430 Representation of an associative algebra
22 KB (3,137 words) - 20:01, 15 March 2023
$#A+1 = 134 n = 1
A [[characteristic class]] defined for complex vector bundles. A Chern class of the complex vector bundle $ \xi $
13 KB (1,910 words) - 19:40, 7 January 2024
''measure of a set''
...ce. The notion of the measure of a set arose in the theory of functions of a real variable in connection with the study and improvement of the notion of
46 KB (7,065 words) - 19:30, 1 January 2021
...rsion units in ${\bf Z} G$. The famous Zassenhaus conjecture says that for a [[Finite group|finite group]] $G$ all torsion units of ${\bf Z} G$ are rati
...d some unit $x \in \mathbf{Q} G$. This conjecture was proved to be true by A. Weiss, first for $p$-groups [[#References|[a16]]] and then for nilpotent g
9 KB (1,457 words) - 17:05, 26 January 2021
A polynomial describing the number of derivative constants in the solution of a system of partial differential equations; it is an analogue of the [[Hilber
be a differential [[Extension of a differential field|extension of a differential field]] $ F $ .
5 KB (651 words) - 08:36, 1 July 2022
in a finite-dimensional vector space $ V $
there is a $ G $ -
6 KB (818 words) - 12:00, 16 December 2019
...ting various objects when these are to be considered as indistinguishable. A basic tool in enumeration problems is provided by generating functions (cf.
...obtaining generating functions is the fact that many discrete objects have a natural order (cf. [[#References|[1]]], [[#References|[5]]]). In what follo
8 KB (1,343 words) - 08:52, 12 November 2023
...$, see [[#References|[a1]]]. For an interplay between the geometry and the algebra, see [[#References|[a2]]].
...rametrization. To obtain the parametrization of the circle, one cuts it by a line of slope $t$ through the point $( - 1,0 )$ and notes that it meets the
16 KB (2,510 words) - 17:43, 1 July 2020
$#A+1 = 30 n = 0
''of a cochain complex $ K ^ { . } = ( K ^ {n} , d _ {n} ) $
3 KB (493 words) - 17:45, 4 June 2020
The notion of a ''group scheme'' is a generalization of the concept of an
schemes over a ground scheme $S$; a
5 KB (831 words) - 21:59, 5 March 2012
..., X _ { n } ]$ is determined by giving a finite set of generators, called a basis. Hermann states explicitly that one can give an upper bound for the n
...mann's contribution lay fallow for decades except mainly for the notice of a few gaps:
10 KB (1,498 words) - 14:47, 3 February 2024
A [[Differential field|differential field]] $ F \supset F _{0} $
with a set $ \Delta $
13 KB (1,981 words) - 20:02, 17 December 2019
$#A+1 = 72 n = 0
$#C+1 = 72 : ~/encyclopedia/old_files/data/V096/V.0906430 Vector field on a manifold
6 KB (884 words) - 08:28, 6 June 2020
$#A+1 = 23 n = 1
A function that does not change under any permutation of its independent vari
5 KB (731 words) - 07:17, 24 March 2023
A real or complex Jordan pair $V = ( V ^ { + } , V ^ { - } )$ (i.e. a pair $V = ( V ^ { + } , V ^ { - } )$ of real or complex vector spaces toget
...nces|[a8]]], [[#References|[a9]]] for so-called $H ^ { * }$-triple theory. A general account of both the geometric and the non-geometric theory of Jorda
14 KB (2,076 words) - 17:00, 1 July 2020
A complex (sometimes, real) number that is a root of a polynomial
...e $n$ is any positive integer, is an algebraic number of degree $n$, being a root of the irreducible polynomial $x^n-2$.
10 KB (1,645 words) - 17:08, 14 February 2020
...$A$. Usually, $A$ is taken to be either the multiplicative group $k^*$ of a field $k$ or the subgroup
...character of a group was originally introduced for finite groups $G$ with $A = T$ (in this case every character $G → ℂ^*$ takes values in $T$).
3 KB (438 words) - 13:36, 23 July 2015
$#A+1 = 150 n = 0
...t theorem establishes a connection between the infinitesimal symmetries of a functional of the form
16 KB (2,336 words) - 08:02, 6 June 2020
...\text{p} }$ is a finite extension of $\widehat { \mathbf{Q} } _ { p }$ or a finite extension of $\mathbf{F} _ { p } ( ( t ) )$ ($\text{p}$ is ultra-met
...hat { K } _ { \operatorname {p} } )$, where $V$ is an affine algebraic set over $\widehat { K } _ { \text{p} }$. In each case, $V ( \widehat { K } _ { \ope
15 KB (2,309 words) - 06:58, 13 February 2024
$#A+1 = 77 n = 0
...operational calculus into algebraic form, using the concept of a function ring. The most general concept of an operational calculus is obtained using gene
12 KB (1,635 words) - 14:54, 7 June 2020
$#A+1 = 37 n = 1
For a positive square-free integer $ d > 0 $,
5 KB (775 words) - 07:59, 26 March 2023
$#A+1 = 88 n = 1
...ordered set]] in which each two-element subset has both a least upper and a greatest lower bound. This implies the existence of such bounds for every n
14 KB (2,066 words) - 15:31, 11 November 2023
$#A+1 = 39 n = 0
Originally, a measurable function was understood to be a function $ f ( x) $
3 KB (434 words) - 08:00, 6 June 2020
$#A+1 = 158 n = 0
$#C+1 = 158 : ~/encyclopedia/old_files/data/T092/T.0902400 Tensor on a vector space
13 KB (1,934 words) - 08:25, 6 June 2020
$#A+1 = 499 n = 0
''of a topological space $ X $''
38 KB (5,928 words) - 19:35, 5 June 2020
...Then there exists an $x \in \widetilde{\mathbf{Z}}$ such that $f ( x )$ is a unit of $\widetilde{\bf Z}$.
...he ideal of $\widetilde{\bf Z}$ generated by its coefficients is the whole ring.
11 KB (1,771 words) - 16:57, 1 July 2020
$#A+1 = 261 n = 2
$#C+1 = 261 : ~/encyclopedia/old_files/data/L058/L.0508360 Lie\AAhadmissible algebra
22 KB (3,154 words) - 19:10, 26 March 2023
$#A+1 = 31 n = 0
A [[Matrix|matrix]] over a commutative ring $ R $
3 KB (437 words) - 14:54, 7 June 2020
$#A+1 = 366 n = 3
...certain cohomology functors into others (most often — into themselves). By a cohomology operation of type $ ( n , m ; \pi , G ) $,
29 KB (4,197 words) - 09:49, 26 March 2023
$#A+1 = 15 n = 0
A locally small [[Full subcategory|full subcategory]] $ \mathfrak S $
3 KB (469 words) - 16:39, 17 March 2023
...the resulting decomposition of $\frak g$, is obtained. A further choice of a positive root system $\Delta ^ { + } \subset \Delta$ determines subalgebras
...unity, a resolution of a left $A$-module $M$ is an exact chain complex of $A$-modules:
12 KB (1,752 words) - 07:39, 10 February 2024
$#A+1 = 181 n = 0
...mological nature that, in contrast to homology, depend contravariantly, as a rule, on the objects of the basic category on which they are defined. In co
16 KB (2,386 words) - 16:47, 20 January 2024
A classical group is the group of automorphisms of some
...example, groups of semi-linear transformations of $E$ preserving $f$ up to a scalar factor and an automorphism of $K$).
12 KB (1,991 words) - 18:05, 29 November 2014
A group having the same constant coefficient homology as the trivial group (c
..., the plus-construction of a topological space is completely determined by a certain perfect, locally free, and hence acyclic, group {{Cite|BeCa}}.
6 KB (873 words) - 08:18, 18 February 2022
...w of [[associativity]]. A semi-group is a generalization of the concept of a [[group]]: only one of the group axioms is retained — associativity; this
...e a self-contained branch of modern algebra with a rich store of problems, a broad range of methods and strong links with many fields of mathematics, bo
17 KB (2,435 words) - 09:18, 2 April 2023
...generalization of the concept of a divisor of an element of a commutative ring. First introduced by E.E. Kummer
...by the divisor $\def\f#1{\mathfrak {#1}} \f a\in D_0$ if $\f a$ divides $(a)$ in $D_0$.
16 KB (2,805 words) - 02:18, 6 January 2022
A [[mapping]] $x\rightarrow\lVert x\rVert$ from a [[vector space]] $X$ over the field of real or complex numbers into the real numbers, subject to the
...mplete in this metric is called a [[Banach space]]. Every normed space has a Banach completion.
8 KB (1,284 words) - 21:49, 6 June 2016
$#A+1 = 31 n = 3
A fundamental theorem in [[K-theory| $ K $-
3 KB (474 words) - 13:07, 24 December 2020
$#A+1 = 387 n = 0
A differential form of degree $ p $, a
27 KB (4,062 words) - 01:31, 7 May 2022
...is small. Often the set $X$ is a topological space and the class $\mathcal{A}$ is related to the topology of $X$.
...dditive and finite on compact sets, then $\mu$ is called (by some authors) a ''regular content'' if
5 KB (751 words) - 09:47, 16 August 2013
...of considering the operators $\hat { A } = A \oplus B$ and $\hat { X } = ( A , B )$ on $\widehat { H } = H \oplus H$, S.K. Berberian [[#References|[a7]]
...a large number of papers, and various generalizations of it have appeared over the past four decades. Broadly speaking, these generalizations fall into th
11 KB (1,613 words) - 07:43, 27 January 2024
...omplex characters on elements of order divisible by $p$, are controlled to a considerable extent by the $p$-local structure. Brauer's three main theorem
...$. The [[Group algebra|group algebra]] $R G$ has a unique decomposition as a direct sum of indecomposable two-sided ideals,
9 KB (1,465 words) - 17:46, 1 July 2020
...ducible factors, essentially uniquely (cf. also [[Factorial ring|Factorial ring]]). For an efficient version of Gauss' theorem, one asks to find these fact
...senhaus [[#References|[a3]]], is to compute $\sigma$ by repeated squaring. A third method (J. von zur Gathen and V. Shoup [[#References|[a8]]]) uses the
12 KB (1,739 words) - 13:11, 26 March 2023
...gful to ask whether they are true or false. For instance, the proposition "a whale is an animal" is true, while the statement "all angles are right angl
...is way depend on the truth or falsehood of the initial propositions and on a corresponding treatment of the connectives as operations on propositions. O
27 KB (3,984 words) - 15:58, 22 December 2019
...u ^ { \Lambda } / \mathcal{I}$ of classical function spaces $\mathcal{V}$. A further approach uses analytic continuation and asymptotic series of distri
...lication of distributions]]) are retained up to this equivalence relation. A list of typical examples of generalized function algebras follows:
12 KB (1,717 words) - 17:43, 1 July 2020
...\operatorname {Mod}$ of left modules over a [[Commutative ring|commutative ring]] $k$ are certain $k$-linear mappings $R : V \otimes _ { k } V \rightarrow
...2 } R _ { 23 } R _ { 12 } = R _ { 23 } R _ { 12 } R _ { 23 }$, then $R$ is a Yang–Baxter operator in $\square _ { k }\operatorname {Mod}$ (see [[#Refe
19 KB (2,895 words) - 17:45, 1 July 2020
$#A+1 = 104 n = 0
...phism of schemes that is separated, universally closed and of finite type. A morphism of schemes $ f : X \rightarrow Y $
9 KB (1,267 words) - 08:08, 6 June 2020
A group whose operation is commutative (cf. [[Commutativity|Commutativity]]).
...$\mathbf{Q}^+$ is Abelian; it is moreover a [[locally cyclic group]], i.e. a group in which all finitely generated subgroups are cyclic. Finally, the gr
11 KB (1,810 words) - 22:12, 29 August 2015
...e desire to find effective algorithms in [[Homological algebra|homological algebra]]. The cornerstone of the theory is an important algorithm which, when conv
...opy between the identity and $\nabla f$, while $f \nabla$ is the identity. A standard notation for an strong deformation retract is the following:
18 KB (2,748 words) - 01:18, 15 February 2024
...a tilting module over $\Gamma ^ { \operatorname{op} }$. Hence $D ( T )$ is a cotilting $\Gamma$-module.
...T$ of $\Lambda$-modules cogenerated by $T$. The category $\mathcal{Y}$ is a torsion-free class and there is an associated torsion pair $\cal ( X , Y )$
16 KB (2,221 words) - 09:47, 11 November 2023
...cifying it are: the Darboux transformation is functionally parametrized by a pair of solutions of the equation and the transform vanishes if the solutio
...he Darboux transformation appears in the paper of Th.-F. Moutard (1875) as a specification of the [[Moutard transformation|Moutard transformation]] [[#R
14 KB (2,023 words) - 17:02, 1 July 2020
...geneous structures), etc. The name finite mathematics is sometimes used as a synonym for discrete mathematics and discrete analysis. In what follows the
...of investigation will depend on the objective of the study and on whether a discrete or continuous model of the phenomenon under study is to be conside
9 KB (1,264 words) - 16:56, 15 April 2012
A ''Lie algebra'' is
...k$-module $L$ over a commutative ring $k$ with a unit that is endowed with a bilinear mapping $(x,y)\mapsto [x,y]$ of $L\times L$ into $L$ having the fo
25 KB (4,037 words) - 07:06, 23 April 2016
...The theory of invariants arose from a number of problems in number theory, algebra and geometry. C.F. Gauss in his studies on the theory of binary quadratic f
...he corresponding transformation groups. The Memoir on hyperdeterminants by A. Cayley (1846) must apparently be regarded as the first work on the theory
22 KB (3,406 words) - 07:08, 6 May 2022
$#A+1 = 61 n = 0
A quaternionic structure on a real vector space $ V $
8 KB (1,179 words) - 15:10, 19 January 2021
...he collection of all first-order sentences true in a given structure forms a complete first-order theory $ T $.
...e algebraically closed fields of characteristic zero. Complete theories in a fixed countable language $ L $
11 KB (1,671 words) - 11:38, 22 December 2019
A commutative group scheme is a group scheme $G$ over a basis scheme $S$, the value of which on any
...[[Abelian scheme|Abelian schemes]] and [[Algebraic torus|algebraic tori]]. A generalization of algebraic tori in the framework of the theory of group sc
4 KB (629 words) - 20:08, 15 December 2020
...an, cf. also [[Abelian difference set|Abelian difference set]]) has led to a satisfactory theory. The special case of cyclic difference sets is the one
where $q$ is a prime power and $d$ a positive integer, include the classical Singer difference sets associated w
9 KB (1,331 words) - 19:36, 13 February 2024
$#A+1 = 159 n = 0
A sequence of differential modules, each of which is the homology module of t
14 KB (2,075 words) - 09:23, 18 February 2022
$#A+1 = 244 n = 1
Historically, the earliest theory of a [[Cohomology of algebras|cohomology of algebras]].
16 KB (2,427 words) - 09:48, 26 March 2023
$#A+1 = 159 n = 1
be a [[Field|field]], let $ V $
11 KB (1,659 words) - 08:43, 26 March 2023
$#A+1 = 103 n = 3
A term used by itself or as a part of standard expressions in a number of similar meanings. The older term, cobordism, is still employed.
12 KB (1,704 words) - 08:36, 26 March 2023
...] $P$ on the group $M$ satisfying two conditions which makes the structure a common generalization of:
1) a [[Normal subgroup|normal subgroup]] of $P$;
9 KB (1,326 words) - 16:58, 1 July 2020
$#A+1 = 101 n = 0
A rational function is a function $ w = R ( z) $,
8 KB (1,257 words) - 03:49, 4 March 2022
...re restricted setting, a skein module is a [[Module|module]] associated to a [[Three-dimensional manifold|three-dimensional manifold]] by considering li
...ensional manifold (e.g. surfaces in a manifold, geometric decomposition of a manifold)?
11 KB (1,644 words) - 16:45, 1 July 2020
$#A+1 = 386 n = 1
''of a set $ X $''
28 KB (4,564 words) - 07:37, 26 March 2023
$#A+1 = 117 n = 0
...uished intermediate fields of algebraic extensions of fields equipped with a [[Valuation|valuation]] are considered. Let $ L \mid K $
7 KB (1,108 words) - 08:09, 6 June 2020
...a unary [[relation symbol]] for being an element of the valuation ring, or a binary relation symbol for valuation divisibility $v(x)\leq v(y)$ (cf. [[St
...t in the proof that Hilbert's 10th problem has a positive solution for the ring of all algebraic integers (see [[Algebraic Diophantine equations|Algebraic
12 KB (1,846 words) - 09:28, 26 November 2016
...amental problems in [[Algebraic geometry|algebraic geometry]] (commutative algebra, polynomial ideal theory) can be reduced by structurally easy algorithms to
...d of Gröbner bases has been carried over to domains other than polynomials over fields, e.g. polynomials with coefficients in certain rings, non-commutativ
6 KB (1,000 words) - 17:05, 7 July 2014
$#A+1 = 160 n = 9
...ic $ K $-theory]], it is sometimes called topological $ K $-theory. In a wide sense, the term "K-theory" is used to denote the branch of mathematics
17 KB (2,459 words) - 07:32, 26 February 2022
...ex $s$ is a simplex, called a face of $s$, and every one-element subset is a simplex.
...ny simplices. A simplicial complex is called ordered if its vertices admit a partial ordering that is linear on every simplex.
11 KB (1,773 words) - 09:56, 13 February 2024
...analysis — the theory of Fourier series and Fourier integrals — underwent
a rapid development, stimulated by physical problems, in the 18th century and
...6/h046420/h0464201.png" /> be
a complex-valued square-summable function on
a circle of unit length (or on the segment <img align="absmiddle" border="0"
66 KB (9,085 words) - 17:28, 31 March 2020
$#A+1 = 299 n = 0
...ith various structures. The algebraic basis of the method is the Grassmann algebra. Let $ V $
21 KB (3,193 words) - 11:01, 4 June 2020
$#A+1 = 24 n = 0
such that for any monomorphism $ \alpha : A ^ \prime \rightarrow A $
4 KB (643 words) - 22:12, 5 June 2020
$#A+1 = 210 n = 0
A non-empty set on which some [[Order (on a set)|order relation]] is given.
21 KB (3,255 words) - 11:54, 19 March 2023
...finitely-generated semi-groups of matrices over a nilpotent or commutative ring, and the finitely-generated regular semi-groups that are nilpotent of class
...The semi-lattice of a family of reductive residually-finite semi-groups is a residually-finite semi-group.
5 KB (643 words) - 16:42, 19 September 2014
$#A+1 = 89 n = 2
...#References|[1]]], Burnside also showed that all groups of order $ p ^ {a} q ^ {b} $,
15 KB (2,211 words) - 08:50, 26 March 2023
$#A+1 = 126 n = 0
$#C+1 = 126 : ~/encyclopedia/old_files/data/A011/A.0101460 Algebraic cycle
12 KB (1,874 words) - 06:08, 17 April 2024
$#A+1 = 313 n = 1
...assification of all possible pairwise non-isomorphic complex structures on a given differentiable real manifold. The fundamental idea (which must be cre
41 KB (5,916 words) - 11:24, 26 March 2023
...ear group as a group of non-singular square $ ( n \times n ) $ -matrices over $ K $ . In this way an isomorphism is established between linear and matr
...]]. Therefore henceforth (unless stated otherwise) only linear groups over a field will be considered.
16 KB (2,362 words) - 18:01, 12 December 2019
...the notion of vertex algebra and of the variant notion of vertex operator algebra are given below.
...y to describe hypothesized interactions of certain elementary particles at a "vertex" (cf. also [[Vertex operator|Vertex operator]]).
20 KB (2,919 words) - 00:57, 15 February 2024
theory one constructs a contravariant functor
theory in algebraic geometry was in its very origin. It was a proof of a generalization (in particular, to smooth varieties of arbitrary dimension)
8 KB (1,120 words) - 20:06, 31 October 2023
$#A+1 = 124 n = 0
A generalization of the notion of a [[Binary relation|binary relation]] (usually) between two sets or mathemati
8 KB (1,265 words) - 17:31, 5 June 2020
$#A+1 = 540 n = 4
...nd related to various structures in the stable tangent or normal bundle to a manifold. Cobordism theory is dual (in the sense of [[S-duality| $ S $-
42 KB (6,290 words) - 19:33, 17 January 2024
A Drinfel'd $A$-module, (where $A$ is an appropriate
[[Ring|ring]]) over a
19 KB (3,204 words) - 20:11, 14 April 2012
$#A+1 = 248 n = 1
...ctors in a conjunctive normal form. In this case a simplest form is called a shortest form. In view of the duality of disjunctive and conjunctive normal
22 KB (3,328 words) - 08:31, 26 March 2023
$#A+1 = 423 n = 1
...re closed with respect to successive composition (or product) of mappings. A category $ \mathfrak C $
43 KB (6,447 words) - 09:17, 26 March 2023
..., in the (numerical) solution of linear equations and eigenvalue problems. A few well-known factorizations are listed below.
...$Q$ and a right-triangular $m\times n$-matrix $R$ such that $A=QR$. Here, a right-triangular $(m\times n)$-matrix, $m\ge n$, is of the form
9 KB (1,294 words) - 22:47, 28 February 2015
$#A+1 = 87 n = 0
...athematics on the other, have determined the development of the concept of a number.
23 KB (3,482 words) - 08:03, 6 June 2020
...ncepts of general topology are the concepts of a [[topological space]] and a [[continuous mapping]], introduced by F. Hausdorff in 1914.
...opological invariant. Proof of the topological invariance of a property of a space is only required when it is formulated with the aid of additional str
10 KB (1,522 words) - 19:35, 25 March 2023
$#A+1 = 298 n = 0
$#C+1 = 298 : ~/encyclopedia/old_files/data/A110/A.1100680 Arguesian lattice,
29 KB (4,201 words) - 16:31, 9 December 2023
$#A+1 = 62 n = 0
A stable [[Cohomology operation|cohomology operation]] $ Sq ^ {i} $,
8 KB (1,093 words) - 08:21, 13 January 2024
$#A+1 = 97 n = 0
$#C+1 = 97 : ~/encyclopedia/old_files/data/A110/A.1100700 Arrangement of hyperplanes
9 KB (1,307 words) - 18:48, 5 April 2020
$#A+1 = 70 n = 3
A generalization of the concept of a commutative [[Formal group|formal group]] of finite height. The homomorphis
6 KB (885 words) - 08:04, 6 June 2020
$#A+1 = 188 n = 2
...xpressing the homology (or cohomology) of a tensor product of complexes or a direct product of spaces in terms of the homology (or cohomology) of the fa
16 KB (2,248 words) - 22:15, 5 June 2020
$#A+1 = 227 n = 0
The problem of describing the integral manifolds of maximal dimension for a [[Pfaffian system|Pfaffian system]] of Pfaffian equations
17 KB (2,624 words) - 19:27, 9 January 2024
...and which is a powerful method for studying many problems in contemporary algebra, geometry, topology, and analysis.
...and the projection $p$ is called the sheaf of Abelian groups (rings, etc.) over $X$ associated with the pre-sheaf $F$.
26 KB (4,342 words) - 15:06, 15 July 2014