On modules over groups

Abstract

For a finite group G, by the endomorphism ring of a module M over a commutative ring R, we define a structure for M to make it an RG-module so that we study the relations between the properties of R-modules and RG-modules. Mainly, we prove that Rad R M is an RG-submodule of M if M is an RGmodule; also Rad R M ⊆ Rad RG M where Rad A M is the intersection of the maximal A-submodule of module M over a ring A. We also verify that M is an injective (projective) R-module if and only if M is an injective (projective) RG-module.

On submodules of modules over group rings

“…The history of group rings dates back to long time and since then, many survey articles have appeared ( [3], [6], [12], [13], [17]). But modules over group rings are one of the subjects studied in recent years by a lot of authors interested in algebra ( [4], [5], [7], [8], [16]). As distinct from the definition of group module over group rings as defined in [7], we gave a definition of group module over group rings with the help of a group homomorphism from G to End(M) in [16].…”

“…But modules over group rings are one of the subjects studied in recent years by a lot of authors interested in algebra ( [4], [5], [7], [8], [16]). As distinct from the definition of group module over group rings as defined in [7], we gave a definition of group module over group rings with the help of a group homomorphism from G to End(M) in [16]. As a continuous study of our previous paper, in here by this previous definition, we give some characterizations for modules over group rings in this paper.…”

“…In here, M is an RG-module with this multiplication and the group homomorphism τ is a representation of G for M over R ( [16]).…”