• Title, Summary, Keyword: one-sided regular element

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ON COMMUTATIVITY OF REGULAR PRODUCTS

  • Kwak, Tai Keun;Lee, Yang;Seo, Yeonsook
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1713-1726
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    • 2018
  • 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.

ON LIFTING OF STABLE RANGE ONE ELEMENTS

  • Altun-Ozarslan, Meltem;Ozcan, Ayse Cigdem
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.793-807
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    • 2020
  • Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly appeared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lifting of elements having (idempotent) stable range one from a quotient of a ring R modulo a two-sided ideal I by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring R is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if R is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.

WEAKLY STABLE CONDITIONS FOR EXCHANGE RINGS

  • Chen, Huanyin
    • Journal of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.903-913
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    • 2007
  • A ring R has weakly stable range one provided that aR+bR=R implies that there exists a $y{\in}R$ such that $a+by{\in}R$ is right or left invertible. We prove, in this paper, that every regular element in an exchange ring having weakly stable range one is the sum of an idempotent and a weak unit. This generalize the corresponding result of one-sided unit-regular ring. Extensions of power comparability and power cancellation are also studied.