Miyashita Action in Strongly Groupoid Graded Rings

We determine the commutant of homogeneous subrings in strongly groupoid graded rings in terms of an action on the ring induced by the grading. Thereby we generalize a classical result of Miyashita from the group graded case to the groupoid graded situation. In the end of the article we exemplify this result. To this end, we show, by an explicit construction, that given a finite groupoid $G$, equipped with a nonidentity morphism $t : d(t) \to c(t)$, there is a strongly $G$-graded ring $R$ with the properties that each $R_s$, for $s \in G$, is nonzero and $R_t$ is a nonfree left $R_{c(t)}$-module.Comment: This article is an improvement of, and hereby a replacement for, version 1 (arXiv:1001.1459v1) entitled "Commutants in Strongly Groupoid Graded Rings

Noncrossed Product Matrix Subrings and Ideals of Graded Rings

We show that if a groupoid graded ring has a certain nonzero ideal property and the principal component of the ring is commutative, then the intersection of a nonzero twosided ideal of the ring with the commutant of the principal component of the ring is nonzero. Furthermore, we show that for a skew groupoid ring with commutative principal component, the principal component is maximal commutative if and only if it is intersected nontrivially by each nonzero ideal of the skew groupoid ring. We also determine the center of strongly groupoid graded rings in terms of an action on the ring induced by the grading. In the end of the article, we show that, given a finite groupoid $G$, which has a nonidentity morphism, there is a ring, strongly graded by $G$, which is not a crossed product over $G$

Commutativity and Ideals in Category Crossed Products

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

Crossed product algebras defined by separable extensions

AbstractWe generalize the classical construction of crossed product algebras defined by finite Galois field extensions to finite separable field extensions. By studying properties of rings graded by groupoids, we are able to calculate the Jacobson radical of these algebras. We use this to determine when the analogous construction of crossed product orders yield Azumaya, maximal, or hereditary orders in a local situation. Thereby we generalize results by Haile, Larson, and Sweedler