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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.
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참고문헌 (8)
참고문헌 신청
D. D. Anderson / 1999 / Semigroups and rings whose zero products commute / Comm. Algebra 27(6):2847~2852
R. Antoine / 2008 / Nilpotent elements and Armendariz rings / J. Algebra 319(88):3128~3140
C. Huh / 2002 / p.p. rings and generalized p.p. rings / J. Pure Appl. Algebra 167(1):37~52
C. Huh / 2004 / Examples of strongly π-regular rings / J. Pure Appl. Algebra 189(1-3):195~210
C. Huh / 2002 / Armendariz rings and semicommutative rings / Comm. Algebra 30(2):751~761
R. Antoine / 2008 / Nilpotent elements and Armendariz rings / J. Algebra 319(88):3128~3140
C. Huh / 2002 / p.p. rings and generalized p.p. rings / J. Pure Appl. Algebra 167(1):37~52
C. Huh / 2004 / Examples of strongly π-regular rings / J. Pure Appl. Algebra 189(1-3):195~210
C. Huh / 2002 / Armendariz rings and semicommutative rings / Comm. Algebra 30(2):751~761
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