Integrality of subrings of matrix rings

Subrings of zero-dimensional rings

When Sarah Glaz, Bill Heinzer and the junior author of this article approached Robert with the idea of editing a book dedicated to his work, we asked him to give us a list of his work and to comment on it to the extent he felt comfortable. As usual, he was extremely thorough in his response. When the authors of this article began to consider what topic we wanted to write about, we were impresse...

Differential Polynomial Rings of Triangular Matrix Rings

Integrality over Fixed Rings

In this paper we prove that if G is a finite group of automorphisms of a ring R, where the order of G is a unit in R, then R is fully integral of degree m(|G|) over the fixed ring R. Here m is a function of the order of G. This gives a positive answer to a well-known question of S. Montgomery and extends the result of D. S. Passman for abelian group actions. Our theorem can be viewed as a gener...

Subrings of Artinian and Noetherian Rings

An easy consequence of this is that a left Noetherian (respectively left Artinian) ring which is finitely generated over its center is right Noetherian (respectively right Artinian). Theorem 1 follows easily from Theorem 2, which gives a partial converse to the following standard fact: IfR C S are rings, and ifQ is an injective R-module, then Hom~(S, Q) is an injective S-module (this follows, f...

Zero-dimensional Subrings of Commutative Rings

We consider various conditions of the sets of zero-dimensional, Artinian, and von Neumann regular subrings of a commutative ring. Section 1 treats questions of existence of such rings, Section 2 deals with the situation in which all subrings belong to one of the three classes, and Section 3 is concerned with the behavior of the sets under intersection. In Section 4 we give a brief survey of som...