On modules over group rings.
Let \(RG\) be the group ring of a group \(G\) over an associative ring \(R\) with identity. Suppose that \(M\) is a unitary right \(R\)-module and denote by \(MG\) the set of all formal finite sums of the form \(\sum_{g\in G}m_gg\), where \(m_g\in M\). In a natural way \(MG\) is a right \(RG\)-module, called the group module of \(G\) by \(M\) over \(RG\). The objective of the paper is to study basic properties of \(MG\) in terms of the properties of \(M\) and \(G\). In particular, the authors prove the module-theoretic versions of several well-known results of group rings, including Maschke's theorem, the classical characterizations of right self-injective group rings and of von Neumann regular group rings. For instance, it is proved that \(MG\) is a semisimple \(RG\)-module if and only if \(M\) is a semisimple \(R\)-module and \(G\) is a finite group whose order is invertible in \(\text{End}_R(M)\). The second main result asserts that \(MG\) is a regular \(RG\)-module if and only if \(M\) is a regular \(R\)-module, \(G\) is locally finite and the order of each finite subgroup of \(G\) is invertible in \(\text{End}_R(M)\). Moreover, \(MG\) is an injective \(RG\)-module if and only if \(M\) is an injective \(R\)-module and \(G\) is a finite group. The paper contains many other interesting results.
scientific article; zbMATH DE number 6306455