On commutativity of regular products
Bull. Korean Math. Soc. 2018 Vol. 55, No. 6, 1713-1726
https://doi.org/10.4134/BKMS.b171018
Published online November 30, 2018
Tai Keun Kwak, Yang Lee, Yeonsook Seo
Daejin University, Daejin University, Pusan National University
Abstract : We study the one-sided regularity of matrices in upper triangular matrix rings in relation with the structure of diagonal entries. We next consider a ring theoretic condition that $ab$ being regular implies $ba$ being also regular for elements $a, b$ in a given ring. Rings with such a condition are said to be {\it commutative at regular product} (simply, {\it CRP} rings). CRP rings are shown to be contained in the class of directly finite rings, and we prove that if $R$ is a directly finite ring that satisfies the descending chain condition for principal right ideals or principal left ideals, then $R$ is CRP. We obtain in particular that the upper triangular matrix rings over commutative rings are CRP.
Keywords : one-sided regular element, regular element, commutative at regular product, directly finite ring, matrix ring
MSC numbers : 16U80, 16S50
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