Groups of module automorphisms of finite rank

Abstract

Let R be any ring, M an R-module and G a subgroup of $\hbox {Aut}_{\mathrm{R}}\hbox {M}$ of finite rank. We compare the R-module structure of certain R-G images of M with that of certain R-G submodules of M. (We are treating M here as an R-G bimodule). For example, if A $=$ [M, G] and $\hbox {B}=\hbox {M/C}_{\mathrm{M}}\hbox {(G)}$, then we prove that as R-modules, A is Artinian if B is Artinian, B is Noetherian if A is Noetherian and hence A has a composition series of finite length if and only if B does.