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    In algebra, a group ring is a free module and a ring constructed from any given ring and any given group. The group ring is a vector space over the given ring, with the group elements as its basis 1. It is defined as the set of all finite linear combinations of elements of the group 2.

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    What is a group ring k(g)?
    1. The group ring k[G] The main idea is that representations of a group G over a field k are “the same” as modules over the group ring k[G]. First I defined both terms. 1.1. Representations of groups. Definition 1.1. A representation of a group G over a field k is defined to be a group homomorphism : G Autk(V ) where V is a vector space over k.

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    What is a ring in physics?
    Rings10.1 Definitions and ExamplesRecall that a group is a set together with a single binary operation, which toget er satisfy a few modest properties. Loosely speaking, a ring is a set together with two binary oper-ations (called addition and multiplication) that are elated via a dis

    Chapter 10 An Introduction to Rings - Dana C. Ernst

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    How much does a group ring know?
    The group ring Z [ G], on the other hand, knows at least as much as k [ G] for every k (which one can obtain by tensoring with k ); this information should let you deduce something about the entries in the character table of G, although I'm not sure exactly what.

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    Is R A commutative ring?
    R is a commutative ring with 1 , 0. Then R is a fie d i↵ its only ideals are 0 and R.Loosely speaking, the previous results say that fields are “like simple groups” (i.e, groups wi h no non-trivial normal subgroups).Corollary 10.51. If R is a field, then every nonzero ring homomorphism from

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