Finite dimensional group rings

Wilkerson, Ralph W.

doi:10.1090/S0002-9939-1973-0318212-8

Finite dimensional group rings
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by Ralph W. Wilkerson PDF

Proc. Amer. Math. Soc. 41 (1973), 10-16 Request permission

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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