Generalized ξ -rings

Abstract

Let RR be a ring with center C(R)C(R). A ring RR is called a ξring if, for any element x∈Rx∈R, there exists an element y∈Ry∈R such that x−x2y∈C(R)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=x2yx=x2y. In order to make up a natural connection of ξξ-rings with the more general class of von Neumann regular rings, that are rings for which x=xyxx=xyx, we introduce here the so-called generalized ξξ-rings as those rings in which x−xyx∈C(R)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).