Commutativity and Ideals in Category Crossed Products

TL;DR

This paper introduces category crossed products to generalize matrix rings and group graded crossed products, analyzing their centers, commutants, and the relationship between maximal commutativity and ideal intersections.

Contribution

It defines category crossed products and explores their algebraic properties, extending existing theories to a broader class of algebraic structures.

Findings

Characterization of the center and commutant of the coefficient ring

Connection between maximal commutativity and ideal intersections

Generalization of matrix rings and group graded crossed products

Abstract

In order to simultaneously generalize matrix rings and group graded crossed products, we introduce category crossed products. For such algebras we describe the center and the commutant of the coefficient ring. We also investigate the connection between on the one hand maximal commutativity of the coefficient ring and on the other hand nonemptyness of intersections of the coefficient ring by nonzero twosided ideals.