...an associative semi-simple ring one understands a [[classical semi-simple ring]].
[[Category:Associative rings and algebras]]
354 bytes (52 words) - 19:41, 19 October 2014
''with respect to a radical $r$''
...semi-simple ring]]; [[Alternative rings and algebras|Alternative rings and algebras]]; [[Jordan algebra|Jordan algebra]]; [[Lie algebra, semi-simple|Lie algebr
1,013 bytes (144 words) - 16:31, 5 August 2014
...image in $R$), then $R$ is isomorphic to the full matrix ring of a [[local ring]]. This holds, in particular, if $J$ is a [[nil ideal]].
<TR><TD valign="top">[1]</TD> <TD valign="top"> N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)</TD></TR>
915 bytes (144 words) - 16:49, 17 October 2014
''right primitive ring''
...t of primitive rings. A [[Simple ring|simple ring]] is either primitive or radical. The primitive rings with non-zero minimal right ideals can be described by
2 KB (275 words) - 18:10, 11 April 2014
''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
...dition 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
''of a ring $A$''
...(A)$ contains no non-zero quasi-regular ideals. The radical was introduced and studied in detail in 1945 by N. Jacobson [[#References|[1]]].
3 KB (444 words) - 06:49, 21 October 2017
$#C+1 = 36 : ~/encyclopedia/old_files/data/Q076/Q.0706670 Quasi\AAhregular radical
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5 KB (764 words) - 08:09, 6 June 2020
...ortant part in the theory of radicals of rings (cf. [[Radical of rings and algebras]]) [[#References|[1]]].
There is the following generalization of the concept of a prime ring. A ring $R$ is said to be '''semi-prime''' if it does not have non-zero nilpotent i
1 KB (246 words) - 18:30, 25 September 2016
...elements of the form $\sigma-1$, $\sigma\in G$, is nilpotent. In the ring of upper-triangular matrices over a field the matrices with 0's along the main
...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
...dempotents [[#References|[3]]]. Simple rings that are radical in the sense of N. Jacobson are known (cf. [[#References|[1]]]). Simple nil rings were cons
...sociative case (cf. [[Alternative rings and algebras|Alternative rings and algebras]]). See also [[Simple algebra|Simple algebra]].
2 KB (318 words) - 09:55, 26 November 2016
...and algebras]]). Right perfect rings are defined similarly. A left perfect ring need not be right perfect.
...t $R$-module satisfies the maximum condition for $n$-generated submodules; and 11) every projective left $R$-module has a decomposition with respect to wh
3 KB (491 words) - 19:59, 30 October 2016
...otient algebra by which is not strongly degenerate is called the Kostrikin radical.
...in aspects of the theory of Lie algebras" , ''Selected problems in algebra and logic'' , Novosibirsk (1973) pp. 142–160 (In Russian)</TD></TR><TR><TD
1 KB (218 words) - 16:32, 1 May 2014
$#C+1 = 29 : ~/encyclopedia/old_files/data/S083/S.0803900 Self\AAhinjective ring,
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4 KB (550 words) - 08:13, 6 June 2020
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be an associative ring with unit, and $ M $
4 KB (674 words) - 07:14, 12 July 2022
...= 157 : ~/encyclopedia/old_files/data/R077/R.0707130 Radical of rings and algebras
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16 KB (2,540 words) - 08:09, 6 June 2020
...not locally nilpotent is a difficult problem; essentially only one example of such an algebra is known (see [[#References|[5]]]).
...the nilpotency indices of the elements are uniformly bounded) over a field of characteristic zero is nilpotent (Higman's theorem); a nil algebra satisfyi
6 KB (1,020 words) - 16:48, 25 March 2023
''group algebra of a group $G$ over a field $K$''
...associative algebra]] 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\
5 KB (872 words) - 22:00, 29 April 2012
...images of an algebraic algebra (of bounded degree) are algebraic algebras (of bounded degree).
...es), [[nil algebra]]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 near-ring is a (not necessarily Abelian) group with respect to addition, and the right distributive property
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