...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