''representation ring''
...the duality theory in terms of block-algebras. In the more general case of groups $G$ of type I, the study of $R$ may be reduced to the study of tensor produ
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...oup|cyclic]]. There exists for each prime number $p$ a quasi-cyclic group, and it is unique up to an isomorphism. This group is isomorphic to the multipli
...ntegers. A quasi-cyclic group is the union of an ascending chain of cyclic groups $C_n$ of orders $p^n$, $n=1,2,\ldots$; more precisely, it is the [[inductiv
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...product of groups|Free product of groups]]), in non-associative algebras, and in Lie algebras.
...variety of algebras: for example, the ring of integers modulo $2$ and the ring of integers modulo $3$ have no free product in the variety of rings with 1.
2 KB (259 words) - 20:56, 30 July 2014
...r a [[unital ring]] $R$ that contains only two submodules: the null module and $M$ itself.
...of it), then the right $R$-module $V$ is irreducible. 4) If $G$ is a group and $k$ is a field, then the [[irreducible representation]]s of $G$ over $K$ ar
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$#C+1 = 44 : ~/encyclopedia/old_files/data/R082/R.0802450 Ring with operators,
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4 KB (634 words) - 07:02, 30 March 2024
...ts of this algebra (see [[Associative rings and algebras|Associative rings and algebras]]) are separable field extensions of $k$ (cf. [[Separable extensio
====Separable algebra over a ring====
2 KB (316 words) - 21:57, 29 November 2014
the group of all $(n\times n)$ invertible matrices over an associative ring (cf.
[[Associative rings and algebras|Associative rings and algebras]]) $K$
3 KB (592 words) - 20:32, 5 December 2014
...ociated the so-called [[Peirce decomposition|Peirce decomposition]] of the ring. For an $n$-ary algebraic relation $\omega$, an element $e$ is said to be a
...ing as an idempotent element of the semi-group of unary operations. In the theory of $R$-modules, the affine operations are those of the form
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$#C+1 = 74 : ~/encyclopedia/old_files/data/C022/C.0202970 Cohen\ANDMacaulay ring,
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7 KB (1,158 words) - 17:45, 4 June 2020
...erator groups (in particular, rings) and for lattices, in contrast to semi-groups, it is always assumed that the partially ordered set of ideals does not con
...and only if there is an element $z$ in $S$ which is divisible on the left and right by any element of $S$ (that is, $z\in xS\cap Sx$ for any $x\in S$); i
6 KB (961 words) - 18:14, 7 May 2016
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''(in group theory)''
4 KB (568 words) - 14:10, 31 December 2020
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A [[module]] $M$ over a [[commutative ring]] $A$
2 KB (308 words) - 18:37, 11 April 2023
...oup. Examples, in particular, are locally nilpotent divisible torsion-free groups.
...ov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>
2 KB (335 words) - 17:07, 30 July 2014
over a ring of Witt vectors $ W (k) $ (
and $ V _{M} $
3 KB (487 words) - 06:58, 21 March 2023
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whose objects are the right unitary modules over an arbitrary associative ring $ R $
4 KB (572 words) - 08:01, 6 June 2020
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be a [[Commutative ring|commutative ring]], let $ A $
6 KB (919 words) - 17:32, 5 June 2020
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of an integral commutative ring $ A $
3 KB (406 words) - 21:43, 3 January 2021
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...ideals (cf. [[Divisorial ideal|Divisorial ideal]]) of a [[Krull ring|Krull ring]] $ A $
5 KB (820 words) - 19:36, 5 June 2020
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An Abelian group associated with an associative ring in the following manner. It was introduced by J.H.C. Whitehead [[#Reference
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the [[Associative rings and algebras|associative algebra]] over $K$ whose elements are all possible fin
...nt, the group ring is often called the group algebra of the group over the ring as well.
5 KB (872 words) - 22:00, 29 April 2012