On commutative invariants for modules over crossed products of minimax nilpotent linear groups

TL;DR

This paper develops a correspondence between modules over crossed products of minimax nilpotent groups and prime ideal classes, enabling the application of commutative algebra methods to module analysis.

Contribution

It introduces a novel correspondence linking modules over crossed products to prime ideal classes, extending group actions and applying commutative algebra techniques.

Findings

Established a correspondence between modules and prime ideal classes.

Extended group actions to prime ideal sets under certain conditions.

Applied commutative algebra methods to module analysis over crossed products.

Abstract

Let $N$ be a minimax nilpotent torsion-free normal subgroup of a soluble group $G$ of finite rank, $R$ be a finitely generated commutative domain and $R*N$ be a crossed product of $R$ and $N$. In the paper we construct a correspondence between an $R*N$-module $W$ and a finite set $M$ of equivalent classes of prime ideals minimal over $Ann_{kA}(W/WI)$, where $kA$ is a group algebra of an abelian minimax group $A$ and $I$ is an appropriative $G$-invariant ideal of $RG$. It is shown that if $Wg \cong W$ for all $ g \in g $ then the action of the group $G$ by conjugations on $N$ can be extended to an action of the group $G$ on the set $M$. The results allow us to apply methods of commutative algebra to the study of $W$.