INDECOMPOSABLE INJECTIVE MODULES AND A THEOREM OF KAPLANSKY1

Abstract

ABSTRACT In an unpublished result dating back to the middle 50's, Kaplansky proved the following: Kaplansky's Theorem. A commutative ring R is von Neumann regular iff every simple R-module is injective is a V-ring), and iff every local ring is a field. In 1972, the author [11] proved a generalization (Theorem 1.2 below), namely that every VNR with Artinian primitive factor rings is a V-ring. The main result of the present paper is related to a question following Corollary 9 of the author's paper [13] (for commutative rings) about the structure of indecomposable injective modules. Theorem. If R is a commutative ring, then R is a VNR, that is, a V-ring, iff every subdirectly irreducible injective R-module E has a skewfield endomorphism ring. The proof devolves into showing that R is VNR, hence a V-ring by Kaplansky's Theorem. The method is showing that every ideal I such that is subdirectly irreducible is prime, and hence by Birkhoff's Theorem, every ideal of R is semiprime, equivalently, idempotent. This implies that R is VNR.