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group ring (Definition)

For any group $ G$, the group ring $ \mathbb{Z}[G]$ is defined to be the ring whose additive group is the abelian group of formal integer linear combinations of elements of $ G$, and whose multiplication operation is defined by multiplication in $ G$, extended $ \mathbb{Z}$-linearly to $ \mathbb{Z}[G]$.

More generally, for any ring $ R$, the group ring of $ G$ over $ R$ is the ring $ R[G]$ whose additive group is the abelian group of formal $ R$-linear combinations of elements of $ G$, i.e.:

$\displaystyle R[G] := \left\{\left. \sum_{i=1}^n r_i g_i\ \right\vert\ r_i \in R,\ g_i \in G\right\}, $
and whose multiplication operation is defined by $ R$-linearly extending the group multiplication operation of $ G$. In the case where $ K$ is a field, the group ring $ K[G]$ is usually called a group algebra.



"group ring" is owned by djao.
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Also defines:  group algebra
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Cross-references: field, combinations, operation, multiplication, linear combinations, integer, abelian group, additive group, ring, group
There are 13 references to this entry.

This is version 4 of group ring, born on 2002-01-23, modified 2002-11-06.
Object id is 1595, canonical name is GroupRing.
Accessed 7128 times total.

Classification:
AMS MSC20C05 (Group theory and generalizations :: Representation theory of groups :: Group rings of finite groups and their modules)
 20C07 (Group theory and generalizations :: Representation theory of groups :: Group rings of infinite groups and their modules)
 16S34 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Group rings , Laurent polynomial rings)

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