On Modules Over Group Rings

Abstract

Let M be a right module over a ring R and let G be a group. The set MG of all formal finite sums of the form ∑ g ∈ Gmgg where mg ∈ M becomes a right module over the group ring RG under addition and scalar multiplication similar to the addition and multiplication of a group ring. In this paper, we study basic properties of the RG-module MG, and characterize module properties of (MG)RG in terms of properties of MR and G. Particularly, we prove the module-theoretic versions of several well-known results on group rings, including Maschke’s Theorem and the classical characterizations of right self-injective group rings and of von Neumann regular group rings.