Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(5): 1257-1272

Published online September 30, 2019 https://doi.org/10.4134/BKMS.b181038

Copyright © The Korean Mathematical Society.

Reversible and pseudo-reversible rings

Juan Huang, Hai-lan Jin, Yang Lee, Zhelin Piao

Yanbian University; Yanbian University; Daejin University; Yanbian University

Abstract

This article concerns the structure of idempotents in reversi\-ble and pseudo-reversible rings in relation with various sorts of ring extensions. It is known that a ring $R$ is reversible if and only if $ab\in I(R)$ for $a, b\in R$ implies $ab=ba$; and a ring $R$ shall be said to be {\it pseudo-reversible} if $0\neq ab\in I(R)$ for $a, b\in R$ implies $ab=ba$, where $I(R)$ is the set of all idempotents in $R$. Pseudo-reversible is seated between reversible and quasi-reversible. It is proved that the reversibility, pseudo-reversibility, and quasi-reversibility are equivalent in Dorroh extensions and direct products. Dorroh extensions are also used to construct several sorts of rings which are necessary in the process.

Keywords: pseudo-reversible ring, reversible ring, Dorroh extension, Abelian ring, quasi-reversible ring, direct product, free algebra, matrix ring, polynomial ring

MSC numbers: 16U80, 16S50, 16S36

Supported by: This article was supported by the National Natural Science Foundation of China(11361063)