Bimodules in group graded rings

TL;DR

This paper introduces the concept of $G$-controlled group graded rings, providing criteria for their characterization and exploring their structure, especially in strongly graded cases, with applications to intermediate subrings and connections to classical results.

Contribution

It defines $G$-controlled rings, establishes necessary and sufficient conditions for their properties, and characterizes strongly $G$-controlled rings, extending classical results to new algebraic contexts.

Findings

Characterization of $G$-controlled rings

Criteria for strongly $G$-controlled rings

Description of intermediate subrings in $G$-controlled strongly graded rings

Abstract

In this article we introduce the notion of a controlled group graded ring. Let $G$ be a group, with identity element $e$, and let $R=\oplus_{g\in G} R_g$ be a unital $G$-graded ring. We say that $R$ is $G$-controlled if there is a one-to-one correspondence between subsets of the group $G$ and (mutually non-isomorphic) $R_e$-bimodules in $R$, given by $G \supseteq H \mapsto \oplus_{h\in H} R_h$. For strongly $G$-graded rings, the property of being $G$-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general $G$-graded ring to be $G$-controlled. We also give a characterization of strongly $G$-graded rings which are $G$-controlled. As an application of our main results we give a description of all intermediate subrings $T$ with $R_e \subseteq T \subseteq R$ of a $G$-controlled strongly $G$-graded ring $R$. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras.