On classification of semigroup rings
Let RG be the group ring of a finite group G over a commutative ring R with identity. If M, N are left RG-progenerators and \(f\in Hom_{RG}(M,N)\), then \(M_ f=Hom_{RG}(N,M)\) is a finitely generated R-module that is an R-algebra subject to the operation \(x\circ y=xfy\). The set \(M_{{\mathcal C}}(R)\) of classes of a certain equivalence on the set of all algebras \(M_ f\) is a commutative monoid with zero subject to the operation induced by \(\otimes_ R\). This modifies the idea used for a similar construction by \textit{F. R. DeMeyer} and \textit{T. J. Ford} [J. Algebra 113, 379-398 (1988; Zbl 0654.13016)]. If R is a field, then \(M_{{\mathcal C}}(R)\) has no nonzero nilpotents and the group of units of \(M_{{\mathcal C}}(R)\) coincides with the Schur group of R. Moreover, a computation of \(M_{{\mathcal C}}(R)\) is given in the case where R is a field of characteristic zero. Reviewer's note: As observed in the paper, because of the sandwich multiplication in \(M_ f\), the monoids \(M_{{\mathcal C}}(R)\) can be used to classify via Munn algebras the semigroup rings of finite completely 0- simple semigroups. However, this does not apply to semigroup rings of arbitrary finite semigroups as the author seems to claim.
scientific article; zbMATH DE number 4118514
scientific article