# ON COMMUTATIVITY OF REGULAR PRODUCTS

• Kwak, Tai Keun (Department of Mathematics Daejin University) ;
• Lee, Yang (Institute of Basic Science Daejin University) ;
• Seo, Yeonsook (Department of Mathematics Pusan National University)
• Accepted : 2018.05.23
• Published : 2018.11.30

#### 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 commutative at regular product (simply, 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.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

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