...ar-algebra(2)|linear algebra]], [[multilinear algebra]], and [[topological algebra]].
...) and [[Non-associative rings and algebras|non-associative algebras]] are algebras in this sense.
788 bytes (106 words) - 11:22, 9 April 2017
====Separable algebra over a field====
...f this algebra (see [[Associative rings and algebras|Associative rings and algebras]]) are separable field extensions of $k$ (cf. [[Separable extension]]).
2 KB (316 words) - 21:57, 29 November 2014
...momorphic images of an algebraic algebra (of bounded degree) are algebraic algebras (of bounded degree).
...nite algebra]]s (in particular, finite-dimensional ones), [[nil algebra]]s and associative [[skew-field]]s with a countable set of generators over an unco
2 KB (305 words) - 16:55, 16 October 2014
...is [[Alternative rings and algebras|alternative]], non-associative and non-commutative.
|valign="top"|{{Ref|Ku}}|| valign="top"| A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
1 KB (138 words) - 08:50, 12 November 2023
''$\Phi$ that is associative and commutative''
...that a non-null subalgebra of a free algebra over a field $\Phi$ is a free algebra.
1 KB (163 words) - 08:20, 12 November 2023
''weighted
algebra''
...//www.encyclopediaofmath.org/legacyimages/b/b110/b110140/b11014017.png" />-
algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/l
8 KB (1,072 words) - 09:47, 11 April 2020
''$M$ over a commutative ring $R$''
...of $M$. For an arbitrary module $M$ the set $\mathrm{Supp}(M)$ is empty if and only if $M=0$.
918 bytes (149 words) - 20:46, 18 October 2014
...ng if and only if it satisfies the maximum condition for left annihilators and does not contain any infinite direct sums of left ideals.
[[Category:Associative rings and algebras]]
952 bytes (138 words) - 19:44, 19 October 2014
...ring $A$ is called the variety generated by the ring $A$. Every variety of rings is generated by a "universal object" of it; this is a free ring of the gi
...bra with a finite number of generators over a field of characteristic 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 more frequently considered case is when $A$ is a [[Banach algebra|Banach algebra]] with respect to the norm
...operators which commute with the operators of symmetric representations of algebras with an involution (the continuous analogue of the [[Schur lemma|Schur lemm
2 KB (291 words) - 16:02, 10 July 2014
Please remove this comment and the {{TEX|auto}} line below,
...iversal algebra]] with two binary operations (addition and multiplication) and with a set $ \Sigma $
4 KB (634 words) - 07:02, 30 March 2024
[[Ring|ring]] in which the equations $ax=b$ and $ya=b$ with $a\ne 0$ are uniquely solvable. In the case of an associative r
...unique solvability of the equations $ax=1$ and $ya=1$ for any $a\ne 0$. A commutative associative skew-field is called a
4 KB (607 words) - 19:53, 5 March 2012
Let $R$ be a [[commutative ring]] [[unital ring|with identity]] and $S$ a commutative ring containing $R$, with the same identity element. An element $s \in S$ i
...mathfrak{p}$ the localization $R_{\mathfrak{p}}$ is an [[integral domain]] and is closed in its field of fractions. In some of the literature a normal rin
2 KB (305 words) - 16:09, 11 September 2016
''of a closed symmetric algebra $R$ of bounded linear operators on a Hilbert space $H$''
...hat $(R\cup E)'=E$. Diagonal algebras are employed in decomposing operator algebras into irreducible ones.
2 KB (264 words) - 11:41, 26 March 2023
...algebras with associative powers over a given field $F$ forms a variety of algebras which, if the characteristic of the field $F$ is zero, is defined by the sy
...identities which defines it is known [[#References|[3]]]. If a commutative algebra $A$ with associative powers of characteristic other than $2$ has an idempot
2 KB (369 words) - 10:19, 16 March 2023
..._{\mathfrak{M}}$ does not have zero divisors. A ring of polynomials over a commutative Rickart ring is a Rickart ring.
...duced in honour of C.E. Rickart, who studied the corresponding property in rings of operators (see [[#References|[1]]]).
2 KB (418 words) - 06:38, 18 October 2017
...dule of finite type over a Bezout ring is a direct sum of a torsion module and a free module.
[[Category:Associative rings and algebras]]
1 KB (190 words) - 19:52, 2 November 2014
...s the Jacobson radical is nilpotent, and the concepts of a nilpotent ideal and a nil ideal coincide. The latter property also holds in a [[Noetherian ring
...ple, in the [[Principal ideal|principal ideal]] generated by $a$. In a non-commutative ring there may by nilpotent elements that are not contained in any nilpoten
3 KB (470 words) - 23:58, 24 November 2018
...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
...where $\widehat R$ is the [[injective hull]] of $R$ as a right $R$-module, and $H = \mathrm{Hom}_R(\widehat R,\widehat R)$ is the endomorphism ring of the
...D valign="top">[1]</TD> <TD valign="top"> J. Lambek, "Lectures on rings and modules" , Blaisdell (1966)</TD></TR>
2 KB (291 words) - 07:26, 29 November 2014
Please remove this comment and the {{TEX|auto}} line below,
A [[Universal algebra|universal algebra]] which is a [[Group|group]] relative to the addition operation $ + $(
4 KB (643 words) - 08:01, 6 June 2020
Please remove this comment and the {{TEX|auto}} line below,
A commutative local ring to which the [[Hensel lemma|Hensel lemma]] applies, or, accordin
5 KB (787 words) - 22:10, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
...mutative algebra|Commutative algebra]]) in which basic commutative algebra and [[Algebraic geometry|algebraic geometry]] can be performed. Zariski categor
4 KB (519 words) - 08:29, 6 June 2020
''group algebra of a group $G$ over a field $K$''
the [[Associative rings and algebras|associative algebra]] over $K$ whose elements are all possible finite sums of the type $\def\a{
5 KB (872 words) - 22:00, 29 April 2012
$#C+1 = 44 : ~/encyclopedia/old_files/data/D033/D.0303680 Division algebra
Please remove this comment and the {{TEX|auto}} line below,
5 KB (706 words) - 09:14, 28 June 2022
$#C+1 = 55 : ~/encyclopedia/old_files/data/M062/M.0602790 Matrix algebra,
Please remove this comment and the {{TEX|auto}} line below,
5 KB (738 words) - 08:00, 6 June 2020
''Associative rings and algebras'' are
rings and algebras with an associative multiplication, i.e., sets with
11 KB (1,726 words) - 20:09, 15 December 2020
...f. also [[PI-algebra]]) if there is a polynomial in the [[free associative algebra]] $\mathbf{Z}\langle x_1,x_2,\ldots\rangle$ which vanishes under all substi
...over a commutative ring satisfies the standard polynomial of degree $2n$, and no polynomial of lower degree.
4 KB (628 words) - 21:20, 11 December 2017
$#C+1 = 47 : ~/encyclopedia/old_files/data/P110/P.1100170 Poisson algebra
Please remove this comment and the {{TEX|auto}} line below,
8 KB (1,172 words) - 19:18, 17 March 2023
$#C+1 = 134 : ~/encyclopedia/old_files/data/I052/I.0502520 Involution algebra,
Please remove this comment and the {{TEX|auto}} line below,
9 KB (1,381 words) - 22:13, 5 June 2020
...algebras]]; [[Jordan algebra|Jordan algebra]]; [[Mal'tsev algebra|Mal'tsev algebra]]).
...ass of all algebras and in the class of all commutative (anti-commutative) algebras. For these classes, too, there holds an imbedding theorem analogous to that
16 KB (2,433 words) - 21:48, 5 January 2016
...inite-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
Please remove this comment and the {{TEX|auto}} line below,
and separable if $ \cap _ {n \in \mathbf Z } M _ {n} = 0 $.
3 KB (480 words) - 16:44, 13 January 2024
...a near-ring is a (not necessarily Abelian) group with respect to addition, and the right distributive property
...nalogue of a ring of matrices. The notions of a sub-near-ring, of an ideal and of a right module over a near-ring are introduced in the usual manner.
5 KB (767 words) - 11:23, 27 October 2014
...is a [[Ring|ring]] $R$ that is a [[Topological space|topological space]], and such that the mappings
and
3 KB (410 words) - 21:45, 23 July 2012
Please remove this comment and the {{TEX|auto}} line below,
...a unit element (cf. [[Associative rings and algebras]]) in which all right and left ideals are principal, i.e. have the form $ aR $
5 KB (880 words) - 19:00, 9 January 2024
$#C+1 = 40 : ~/encyclopedia/old_files/data/S086/S.0806210 Space over an algebra
Please remove this comment and the {{TEX|auto}} line below,
8 KB (1,172 words) - 16:55, 7 June 2020
Please remove this comment and the {{TEX|auto}} line below,
A theorem that describes all finite-dimensional associative real algebras without divisors of zero; it was proved by G. Frobenius [[#References|[1]]]
5 KB (790 words) - 19:40, 5 June 2020
$#C+1 = 70 : ~/encyclopedia/old_files/data/D033/D.0303860 Double and dual numbers
Please remove this comment and the {{TEX|auto}} line below,
6 KB (895 words) - 19:36, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
be a commutative ring with unit, and let $ A $
4 KB (663 words) - 08:28, 20 January 2024
$#C+1 = 133 : ~/encyclopedia/old_files/data/C021/C.0201040 Cayley\ANDDickson algebra
Please remove this comment and the {{TEX|auto}} line below,
12 KB (1,483 words) - 16:43, 4 June 2020
Please remove this comment and the {{TEX|auto}} line below,
...r $ A $ (cf. also [[Associative rings and algebras|Associative rings and algebras]]). The elements of $ M ( A ) $
5 KB (722 words) - 18:41, 5 October 2023
..._n$ or $M_n(R)$. Throughout this article $R$ is an [[Associative rings and algebras|associative ring]] [[unital ring|with identity]].
...n $R_n$. An associative ring $A$ with identity 1 is isomorphic to $R_n$ if and only if there is in $A$ a set of $n^2$ elements $e_{ij}$, $i,j=1,\ldots,n$,
4 KB (631 words) - 06:20, 13 September 2016
A [[triple system]] closely related to [[Jordan algebra]]s.
...mapping]] $V \times V \times V \rightarrow V$, called a ''triple product'' and usually denoted by $\{ \cdot , \cdot , \cdot \}$ (sometimes dropping the co
3 KB (525 words) - 10:13, 30 April 2018
Please remove this comment and the {{TEX|auto}} line below,
...algebraic structure. The concept of an ideal first arose in the theory of rings. The name ideal derives from the concept of an [[Ideal number|ideal number]
13 KB (2,163 words) - 22:11, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
An algebra over a field for which certain polynomial identities are true.
15 KB (2,252 words) - 08:04, 6 June 2020
Please remove this comment and the {{TEX|auto}} line below,
...ng with a unit, cf. [[Associative rings and algebras|Associative rings and algebras]]). A multilinear form is also called a multilinear function ( $ n $-
6 KB (855 words) - 20:46, 16 January 2024
...= 157 : ~/encyclopedia/old_files/data/R077/R.0707130 Radical of rings and algebras
Please remove this comment and the {{TEX|auto}} line below,
16 KB (2,540 words) - 08:09, 6 June 2020
Please remove this comment and the {{TEX|auto}} line below,
''of a Banach algebra''
3 KB (528 words) - 08:22, 6 June 2020
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...igma$ of $R$ extends uniquely to an automorphism $\hat { \sigma }$ of $Q$, and one says that $\sigma$ is $X$-inner if $\hat { \sigma }$ is inner on $Q$ (c
9 KB (1,402 words) - 13:37, 25 November 2023
...with the Hall element $t \in H$ is then computed in the [[free associative algebra]] (i.e. the ring of polynomials with non-commuting indeterminates $\{a_1,a_
This basis of the free Lie algebra is called the Hall basis (corresponding to the given Hall set). Sometimes t
3 KB (577 words) - 13:35, 20 March 2023
Please remove this comment and the {{TEX|auto}} line below,
and satisfies the relation usually referred to as the [[Leibniz rule]]
6 KB (945 words) - 17:32, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
...ory|category]] of (graded) co-commutative co-algebras (cf. [[Co-algebra|Co-algebra]]). Such an object consists, first, of a sequence $ \{ H _ {i} \} $
7 KB (1,053 words) - 22:11, 5 June 2020
$#C+1 = 89 : ~/encyclopedia/old_files/data/G043/G.0403180 Galois theory of rings
Please remove this comment and the {{TEX|auto}} line below,
6 KB (975 words) - 04:16, 19 March 2022
Please remove this comment and the {{TEX|auto}} line below,
...ry of uniform algebras)]]; [[Commutative Banach algebra|Commutative Banach algebra]]). Much of the content of this work can be found in [[#References|[a3]]].
5 KB (677 words) - 13:07, 7 April 2023
Please remove this comment and the {{TEX|auto}} line below,
...ring makes it possible to take the most important properties of geometric rings axiomatically into account.
4 KB (643 words) - 19:38, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
and $ ax \leq ay $
3 KB (507 words) - 19:38, 5 June 2020
...urier transform does not vanish on the group of characters $\hat G$ of $G$ and if $y$ is a function in $L^\infty(G)$ such that the convolution $(x*y)$ ten
...y compact group, and on the possibility of [[spectral synthesis]] in group algebras for closed ideals belonging to only a finite number of regular maximal idea
2 KB (322 words) - 18:46, 13 April 2017
$#C+1 = 48 : ~/encyclopedia/old_files/data/G044/G.0404660 Graded algebra
Please remove this comment and the {{TEX|auto}} line below,
4 KB (576 words) - 19:42, 5 June 2020
''restricted Lie algebra''
An algebra $ L $
10 KB (1,443 words) - 10:14, 17 December 2019
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
''Leibniz algebra''
9 KB (1,452 words) - 06:17, 15 February 2024
$#C+1 = 92 : ~/encyclopedia/old_files/data/R082/R.0802480 Rings and algebras
Please remove this comment and the {{TEX|auto}} line below,
21 KB (3,225 words) - 09:25, 13 July 2022
$#C+1 = 45 : ~/encyclopedia/old_files/data/G045/G.0405230 Group algebra of a locally compact group
Please remove this comment and the {{TEX|auto}} line below,
6 KB (888 words) - 10:16, 8 May 2022
Please remove this comment and the {{TEX|auto}} line below,
A [[Commutative ring|commutative ring]] with a unit that has a unique maximal ideal. If $ A $
9 KB (1,456 words) - 22:17, 5 June 2020
Please remove this comment and the {{TEX|auto}} line below,
cf. [[Associative rings and algebras|Associative rings and algebras]]). The following properties are equivalent: a) $ R $
14 KB (2,228 words) - 08:10, 6 June 2020
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
''abstract von Neumann algebra''
14 KB (2,118 words) - 16:46, 1 July 2020
[[Associative rings and algebras|Associative rings and algebras]]) $K$
[[Centre of a ring|Centre of a ring]]) $K$. When $K$ is commutative
3 KB (592 words) - 20:32, 5 December 2014
Please remove this comment and the {{TEX|auto}} line below,
be an associative ring with unit, and $ M $
4 KB (674 words) - 07:14, 12 July 2022
Please remove this comment and the {{TEX|auto}} line below,
''of a universal algebra $ A $''
7 KB (1,037 words) - 08:24, 6 June 2020
Please remove this comment and the {{TEX|auto}} line below,
...ional commutative and associative binary operation, called multiplication (and denoted by $ \cdot $)
8 KB (1,218 words) - 08:02, 6 June 2020
Please remove this comment and the {{TEX|auto}} line below,
of modules and any homomorphism $ \beta : P \rightarrow C $
7 KB (1,080 words) - 08:08, 6 June 2020
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
...cap A ^ { \prime }$. This is easily seen to yield an equivalence relation, and the set $Q_{\text{l}} ( R )$ of all equivalence classes $[ A , f ]$ is a ri
10 KB (1,646 words) - 19:47, 23 December 2023
''computer algebra system''
...among the widely available systems is REDUCE, which runs on many platforms and is still being further developed.
9 KB (1,292 words) - 20:51, 18 September 2016
Please remove this comment and the {{TEX|auto}} line below,
\textrm{ and } \ \mathfrak Z _ {r} ( \mathfrak Z _ {l} ( H) ) = H
7 KB (1,087 words) - 08:09, 6 June 2020
...One defines the [[Rank of an algebraic group|rank of an algebraic group]] and the [[Rank of a Lie group|rank of a Lie group]] in a special way.
...k in the theory of Lie algebras (see [[Rank of a Lie algebra|Rank of a Lie algebra]]).
4 KB (631 words) - 10:02, 15 April 2014
[[Universal algebra|universal algebra]] that is a topological space such that the algebraic operations are contin
...e ring $R$ that is a topological space in which the operations of addition and multiplication, as well as the mapping $R\times A\to A$ ($(r,a)\to ra$), ar
9 KB (1,142 words) - 17:15, 15 November 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
A [[Topological algebra|topological algebra]] $A$ over the field of complex numbers whose topology is
...ing separately continuous for both factors. A Banach algebra is said to be commutative if
14 KB (2,346 words) - 22:48, 29 November 2014
Please remove this comment and the {{TEX|auto}} line below,
is called effective if its elements and ring operations can be described effectively as well as the problem of find
10 KB (1,471 words) - 14:52, 30 May 2020
Please remove this comment and the {{TEX|auto}} line below,
...role in ring theory. Every ring with identity has maximal left (also right and two-sided) ideals. The quotient module $ M = R / I $
7 KB (1,160 words) - 08:00, 6 June 2020
...groups has numerous applications in algebraic geometry, class field theory and cobordism theory.
be the functor that associates with an algebra $ B $
17 KB (2,537 words) - 22:38, 15 December 2019
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...d by B.Yu. Weisfeiler and A.A. Leman (cf. also [[Cellular algebra|Cellular algebra]]).
7 KB (1,092 words) - 17:44, 1 July 2020
$#C+1 = 204 : ~/encyclopedia/old_files/data/L058/L.0508430 Lie algebra, graded
Please remove this comment and the {{TEX|auto}} line below,
20 KB (2,828 words) - 19:11, 26 March 2023
Please remove this comment and the {{TEX|auto}} line below,
...numbers. Complex numbers are depicted geometrically by points in the plane and operations on them correspond to the simplest geometric transformations of
11 KB (1,563 words) - 14:54, 7 June 2020
...elation $\norm{x^* x} = \norm{x}^2$ for any element $x \in A$. $\Cstar\!$-algebras were introduced in 1943
...under the name of $B^*$-algebras. The most important examples of $\Cstar$-algebras are:
15 KB (2,316 words) - 21:05, 20 April 2012
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
of the algebra $ {\mathcal B} ( H) $
17 KB (2,631 words) - 19:37, 19 January 2024
Please remove this comment and the {{TEX|auto}} line below,
...cts (injective resolutions). This makes it possible to develop homological algebra in these categories.
4 KB (643 words) - 22:12, 5 June 2020
$#C+1 = 1 : ~/encyclopedia/old_files/data/A011/A.0101340 Algebra
Please remove this comment and the {{TEX|auto}} line below,
17 KB (2,478 words) - 16:09, 1 April 2020
...uced by structurally easy algorithms to the construction of Gröbner bases; and
...le in all modern mathematical software systems, as for example Mathematica and Maple.
6 KB (1,000 words) - 17:05, 7 July 2014
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
...ing|Noetherian ring]]) with [[Maximal ideal|maximal ideal]] $\mathfrak{m}$ and $d = \operatorname { dim } A$. Let $M$ be a finitely-generated $A$-module w
27 KB (4,003 words) - 17:43, 1 July 2020
...tensor product of two unitary modules $V_1$ and $V_2$ over an associative commutative ring $A$ with a unit is the $A$-module $V_1 \tensor_A V_2$ together with an
...product is uniquely defined up to a natural isomorphism. It always exists and can be constructed as the quotient module of the free $A$-module $F$ genera
11 KB (1,992 words) - 03:52, 23 July 2018
$#C+1 = 261 : ~/encyclopedia/old_files/data/L058/L.0508360 Lie\AAhadmissible algebra
Please remove this comment and the {{TEX|auto}} line below,
22 KB (3,154 words) - 19:10, 26 March 2023
$#C+1 = 71 : ~/encyclopedia/old_files/data/E037/E.0307080 Exterior algebra,
Please remove this comment and the {{TEX|auto}} line below,
7 KB (1,013 words) - 19:38, 5 June 2020
$#C+1 = 230 : ~/encyclopedia/old_files/data/C023/C.0203110 Cohomology of algebras
Please remove this comment and the {{TEX|auto}} line below,
19 KB (2,870 words) - 09:48, 26 March 2023
The branch of algebra whose main study is derived functors on various categories of algebraic obj
...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
Please remove this comment and the {{TEX|auto}} line below,
systems closed with respect to subsystems, homomorphic images and Cartesian products.
10 KB (1,438 words) - 16:10, 1 April 2020
Please remove this comment and the {{TEX|auto}} line below,
...tely reducible (cf. [[Reducible representation|Reducible representation]]) and defined over $ \mathbf Q $ (in other words, irreducible finite-dimensiona
18 KB (2,500 words) - 13:04, 18 February 2022
...e can be considered as a right $A$-module and the distinction between left and right modules disappears. Below only left $A$-modules are discussed.
...ts of the theory of modules were connected with the application of methods and ideas of the theory of categories (cf.
23 KB (3,918 words) - 04:31, 23 July 2018
Please remove this comment and the {{TEX|auto}} line below,
which is closed with respect to the operations of multiplication and complex conjugation.
7 KB (1,019 words) - 20:29, 16 January 2024
...all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
''braided Hopf algebra, braided bi-algebra''
21 KB (3,130 words) - 17:42, 1 July 2020