DefinePK

Abstract

Let R be a ring with center C(R). A ring R is called a ξring if, for any element x ∈ R, there exists an element y ∈ R such that x − x2y ∈ C(R). In Proc. Japan Acad. Sci., Ser. A – Math. (1957), Utumi describes the structure of these rings as a natural generalization of the classical strongly regular rings, that are rings for which x = x2 y. In order to make up a natural connection of ξ-rings with the more general class of von Neumann regular rings, that are rings for which x =xyx, we introduce here the so-called generalized ξ-rings as those rings in which x − xyx ∈ C(R). Several characteristic properties of this newly defined class are proved, which extend the corresponding ones established by Utumi in these Proceedings (1957).