Artinian-finitary groups over commutative rings

Abstract

Let $M$ be a module over the commutative ring $R$. We consider the group $G$ of all automorphisms $g$ of $M$ for which $M(g-1)$ is $R$-Artinian. We show that $G$ has a locally residually nilpotent normal subgroup modulo which $G$ is a subdirect product of finitary linear groups over field images of $R$. This can be used to study certain subgroups of $G$. For example, if $H$ is a locally finite subgroup of $G$, then $H$ is isomorphic to a finitary linear group of characteristic zero if $R$ is an algebra over the rationals and $H/O_p(H)$ is isomorphic to a finitary linear group of characteristic the prime $p$ if R has characteristic a power of $p$. It also gives information about $\operatorname{Aut}_RM$ if $M$ itself is $R$-Artinian.