Bulletin of the
Korean Mathematical Society
BKMS

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

Article

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Bull. Korean Math. Soc. 2014; 51(3): 813-822

Printed May 31, 2014

https://doi.org/10.4134/BKMS.2014.51.3.813

Copyright © The Korean Mathematical Society.

Quasipolar matrix rings over local rings

Jian Cui and Xiaobin Yin

Anhui Normal University, Anhui Normal University

Abstract

A ring $R$ is called quasipolar if for every $a\in R$ there exists $p^2=p\in R$ such that $p\in{\rm comm}^2_R(a)$, $a+p\in U(R)$ and $ap\in R^{\rm qnil}.$ The class of quasipolar rings lies properly between the class of strongly $\pi$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2\times 2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2\times 2$ matrix ring to be quasipolar are obtained.

Keywords: quasipolar ring, matrix ring, strongly clean ring, local ring

MSC numbers: 16U99