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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Finite dimensional group rings


Author: Ralph W. Wilkerson
Journal: Proc. Amer. Math. Soc. 41 (1973), 10-16
MSC: Primary 16A26; Secondary 20C05
MathSciNet review: 0318212
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Abstract: A ring is right finite dimensional if it contains no infinite direct sum of right ideals. We prove that if a group $ G$ is finite, free abelian, or finitely generated abelian, then a ring $ R$ is right finite dimensional if and only if the group ring RG is right finite dimensional. A ring $ R$ is a self-injective cogenerator ring if $ {R_R}$ is injective and $ {R_R}$ is a cogenerator in the category of unital right $ R$-modules; this means that each right unital $ R$-module can be embedded in a direct product of copies of $ R$. Let $ G$ be a finite group where the order of $ G$ is a unit in $ R$. Then the group ring RG is a selfinjective cogenerator ring if and only if $ R$ is a self-injective cogenerator ring. Additional applications are given.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0318212-8
Keywords: Group ring, injective, order, cogenerator, rationally closed, dense right ideal, complete ring of quotients, right finite dimensional
Article copyright: © Copyright 1973 American Mathematical Society