Indecomposable injective modules and a theorem of Kaplansky

In an unpublished result dating back to the middle 50's, Kaplansky proved the following: Kaplansky's Theorem.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 (= VNR) iff every simple R-module is injective (= R is a V-ring), and iff every local ring R-m is a field. In 1972, the author (F[72]) 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 (F[74]) (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 R/I is subdirectly irreducible is prime, and hence by Birkhoffs Theorem, every ideal of R is semiprime, equivalently, idempotent. This implies that R is VNR.