The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras

Abstract

Given a partial action $\alpha$ of a groupoid $G$ on a ring $R$, we study the associated partial skew groupoid ring $R \rtimes_{\alpha} G$, which carries a natural $G$-grading. We show that there is a one-to-one correspondence between the $G$-invariant ideals of $R$ and the graded ideals of the $G$-graded ring $R \rtimes_{\alpha}G.$ We provide sufficient conditions for primeness, and necessary and sufficient conditions for simplicity of $R \rtimes_{\alpha}G.$ We show that every ideal of $R \rtimes_{\alpha}G$ is graded if, and only if, $\alpha$ has the residual intersection property. Furthermore, if $\alpha$ is induced by a topological partial action $\theta$, then we prove that minimality of $\theta$ is equivalent to $G$-simplicity of $R$, topological transitivity of $\theta$ is equivalent to $G$-primeness of $R$, and topological freeness of $\theta$ on every closed invariant subset of the underlying topological space is equivalent to $\alpha$ having the residual intersection property. As an application, we characterize condition (K) for an ultragraph in terms of topological properties of the associated partial action and in terms of algebraic properties of the associated ultragraph algebra.