Journal of Prime Research in Mathematics
Vol. 14 (2018), Issue 1, pp. 13 – 17
ISSN: 1817-3462 (Online) 1818-5495 (Print)
ISSN: 1817-3462 (Online) 1818-5495 (Print)
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
Copyright © 2018 Peter V. Danchev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Published: December, 2018.
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.