On soluble groups of module automorphisms of finite rank

Wehrfritz, Bertram A. F.

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 3, pp. 809-818.

Résumé

Let $R$ be a commutative ring, $M$ an $R$-module and $G$ a group of $R$-automorphisms of $M$, usually with some sort of rank restriction on $G$. We study the transfer of hypotheses between $M/C_{M}(G)$ and $[M,G]$ such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose $[M,G]$ is $R$-Noetherian. If $G$ has finite rank, then $M/C_{M}(G)$ also is $R$-Noetherian. Further, if $[M,G]$ is $R$-Noetherian and if only certain abelian sections of $G$ have finite rank, then $G$ has finite rank and is soluble-by-finite. If $M/C_{M}(G)$ is $R$-Noetherian and $G$ has finite rank, then $[M,G]$ need not be $R$-Noetherian.

Détail

DOI : 10.21136/CMJ.2017.0193-16

Classification : 13E05, 20C07, 20F16, 20H99
Keywords: soluble group; finite rank; module automorphisms; Noetherian module over commutative ring