Simple group graded rings and maximal commutativity

Simple item page
dc.creatorÖinert, Johan
dc.date2009-04-29
dc.date.accessioned2026-07-07T13:09:59Z
dc.date.available2026-07-07T13:09:59Z
dc.descriptionIn 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 $R_e \rtimes_σ G$, 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 $R_e \rtimes_σ G$. 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.
dc.identifierhttps://arxiv.org/abs/0904.4661
dc.identifierhttp://arxiv.org/abs/0904.4661
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/228920
dc.subjectRings and Algebras
dc.subject13A02; 16S35
dc.titleSimple group graded rings and maximal commutativity
dc.typetext

Files

Collections

arXiv