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 = K^G 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 A^K 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 = Q_max(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 = K^G is a Galois subring, then A = Q ^r_max (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).