THE IDEAL STRUCTURE OF PARTIAL SKEW GROUPOID RINGS WITH APPLICATIONS TO TOPOLOGICAL DYNAMICS AND ULTRAGRAPH ALGEBRAS

Given a partial action α of a groupoid G on a ring R, we study the associated partial skew groupoid ring with its natural G-grading. In this talk, we will characterize primeness, simplicity, and the graded ideals of partial skew groupoid rings. Furthermore, if α is induced by a topological partial action, we will present the connections between algebraic and topological properties. As an application, we will characterize the condition (K) for an ultragraph in terms of the topological properties of the associated partial action and the algebraic properties of the corresponding ultragraph algebra.