...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