Algebraic Extensions of Commutative Regular Rings

Abstract

In this paper we study algebraic closures for commutative semiprime rings. The main interest, however, is with rings which are regular in the sense of von Neumann. These play the same role with respect to semiprime rings as fields do with respect to integral domains. Two generally distinct notions are defined: “algebraic” and “weak-algebraic” extensions. Each has the transitivity property and yields a closure which is unique up to isomorphism and is “universal”. Both coincide in fields. The extensions here called “algebraic” were studied independently by Enochs [ 5 ] and myself. Our results on these extensions proceed from a different point of view, and allow us to answer a question posed by Enochs. Furthermore, these results are required (and were developed) in order to obtain the weak-algebraic closure, which was the original closure sought. The motivation for the weak-algebraic extensions is found in the work of Shoda [ 14 , p. 134, no. 1].