Subrings of self-injective and FPF rings

Abstract

We say that a ring K is (right) split by a subring A provided that A is an (right) A-module direct summand of K. Then K is said to be a split extension of A. By a theorem of Azumaya [1], a necessary and sufficient condition for this to happen is that K generates the category mod-A of all right A-modules. A classical example of this occurs when A = KG is a Galois subring corresponding to a finite group of invertible order |G|. In order that A be a right self-injective subring of K it is necessary that A split in K, and the latter condition is sufficient for a right self-injective left A-flat extension K of A (Theorem 1).We also study when the (F)PF property is inherited by a subring A: K is right (F)PF if each (finitely generated) faithful right K module generates mod-K. Any quasi-frobenius (QF) ring is right and left PF; any commutative Prufer domain, and any commutative self-injective ring is FPF [4,5].The main theorem on FPF rings states that A inherits the right (F)PF hypothesis on K when K is left faithfully flat right projective generator over A. Now another theorem of Azumaya [1] states that if A is commutative, then any finitely generated faithful projective A-module generates mod-A, hence a corollary is that K FPF => A FPF whenever K is finitely generated projective over a commutative subring A.We apply the foregoing results to a subring A of a right self-injective ring K in the case that A is right non-singular. Then, assuming that AK is flat, by the structure theory of nonsingular rings K (being injective over A on the right) contains a unique injective hull of A which is canonically the maximal quotient ring Q = Qmax(A), and, moreover, then Q splits in K (Theorem 4.) This holds in particular if A is a von Neumann regular ring (Corollary 5). Furthermore, if A = KG is a Galois subring, then A = Q rmax (A) is right self-injective (Theorem 6 and Corollary 7).As a final application we derive a theorem of Armendariz-Steinberg [19] stating that if K is a right self-injective regular ring then the center of K is self-injective (Theorem 10).KeywordsCommutative RingRegular RingQuotient RingInjective HullSplit ExtensionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.