Journal of Prime Research in Mathematics

# Generalized $$\xi$$-rings

Peter V. Danchev
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences ”Acad. G. Bonchev” str., bl. 8, 1113 Sofia, Bulgaria.

$$^{1}$$Corresponding Author: danchev@math.bas.bg

### 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 $$x − x^2y ∈ C(R)$$. 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 $$x = x^2 y$$. 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).

#### Keywords:

Idempotents, nipotents, regular rings, strongly regular rings, $$ξ$$-rings.