• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.945-956
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• 2020

We introduce the concept of S-exchange rings to unify various subclass of exchange rings, where S is a subset of the ring. Many properties on S-exchange rings are obtained. For instance, we show that a ring R is clean if and only if R is left U(R)-exchange, a ring R is nil clean if and only if R is left (N(R) - 1)-exchange, and that a ring R is J-clean if and only if R is left (J(R) - 1)-exchange. As a conclusion, we obtain a sufficient condition such that clean (nil clean property, respectively) can pass to corners and reprove that J-clean passes to corners by a different way.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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• pp.295-305
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• 2005

In this paper, we investigate exchange ideals and get some new characterization of exchange rings. It is shown that an ideal I of a ring R is an exchange ideal if and only if so is $QM_2$(I). Also we observe that every exchange ideal can be characterized by exchange elements.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.2
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• pp.19-26
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• 2001

All rings are assumed to be associative but do not necessarily have an identity. In this paper, we carry out a study of ring theoretic properties of formal triangular matrix rings. Some results are obtained on these rings concerning properties such as being $I_0$-ring, I-ring, exchange ring.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.455-468
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• 2011

We introduce the notion of refinement rings, and generalize some comparability properties of regular rings to the setting of refinement rings.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.873-884
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• 2009

We characterize QB-ideals of exchange rings by means of quasi-invertible elements and annihilators. Further, we prove that every $2\times2$ matrix over such ideals of a regular ring admits a diagonal reduction by quasi-inverse matrices. Prime exchange QB-rings are studied as well.

• Journal of the Korean Mathematical Society
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• v.47 no.4
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• pp.819-830
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• 2010

We prove, in this article, that a ring R is a stable exchange ring if and only if so are all its Pierce stalks. If every Pierce stalks of R is artinian, then $1_R$ = u + $\upsilon$ with u, $\upsilon$ $\in$ U(R) if and only if for any a $\in$ R, there exist u, $\upsilon$ $\in$ U(R) such that a = u + $\upsilon$. Furthermore, there exists u $\in$ U(R) such that $1_R\;{\pm}\;u\;\in\;U(R)$ if and only if for any a $\in$ R, there exists u $\in$ U(R) such that $a\;{\pm}\;u\;\in\;U(R)$. We will give analogues to normal exchange rings. The root properties of such exchange rings are also obtained.

• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.741-756
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• 2008

In this paper, we introduce a new class of rings, SB-rings. We establish various properties of this concept. These shows that, in several respects, SB-rings behave like rings satisfying unit 1-stable range. We will give necessary and sufficient conditions under which a semilocal ring is a SB-ring. Furthermore, we extend these results to exchange rings with all primitive factors artinian. For such rings, we observe that the concept of the SB-ring coincides with Goodearl-Menal condition. These also generalize the results of Huh et al., Yu and the author on rings generated by their units.

• Kyungpook Mathematical Journal
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• v.45 no.4
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• pp.519-526
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• 2005

In this paper, we prove that every exchange ring can be characterized by potent elements. Also we extend [10, Theorem 3.1 and Theorem 4.1] to quasi-clean rings in which every element is a sum of a potent element and a unit.

• 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.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.171-175
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• 2005

In this paper, we introduce the concept of op-idempotents. It is shown that every exchange ring can be characterized by op-idempotents