Simple group graded rings and maximal commutativity

TL;DR

This paper characterizes when strongly group graded rings are simple, linking simplicity to properties of the neutral component and the action of the grading group, with applications to crossed products and dynamical systems.

Contribution

It provides necessary and sufficient conditions for simplicity of strongly group graded rings, especially when the neutral component is maximal commutative, extending to crossed products and dynamical systems.

Findings

Strongly group graded rings are simple iff the neutral component is G-simple when maximal commutative.

Skew group rings are simple iff the neutral component is G-simple and maximal commutative.

Dynamical system-based crossed products are simple under specific dynamical and algebraic conditions.

Abstract

In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring $R = \bigoplus_{g\in G} R_g$ the grading group $G$ acts, in a natural way, as automorphisms of the commutant of the neutral component subring $R_e$ in $R$ and of the center of $R_e$. We show that if $R$ is a strongly $G$-graded ring where $R_e$ is maximal commutative in $R$, then $R$ is a simple ring if and only if $R_e$ is $G$-simple (i.e. there are no nontrivial $G$-invariant ideals). We also show that if $R_e$ is commutative (not necessarily maximal commutative) and the commutant of $R_e$ is $G$-simple, then $R$ is a simple ring. These results apply to $G$-crossed products in particular. A skew group ring $R_e \rtimes_{\sigma} G$, where $R_e$ is commutative, is shown to be a simple ring if and only if $R_e$ is $G$-simple and maximal commutative in $R_e \rtimes_{\sigma} G$. As an interesting example we consider the skew group algebra $C(X) \rtimes_{\tilde{h}} \mathbb{Z}$ associated to a topological dynamical system $(X,h)$. We obtain necessary and sufficient conditions for simplicity of $C(X) \rtimes_{\tilde{h}} \mathbb{Z}$ with respect to the dynamics of the dynamical system $(X,h)$, but also with respect to algebraic properties of $C(X) \rtimes_{\tilde{h}} \mathbb{Z}$. Furthermore, we show that for any strongly $G$-graded ring $R$ each nonzero ideal of $R$ has a nonzero intersection with the commutant of the center of the neutral component.