• Kyungpook Mathematical Journal
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• 제54권1호
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• pp.65-72
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• 2014

A ring R is called weakly semicommutative ring if for any a, $b{\in}R^*$ = R\{0} with ab = 0, there exists $n{\geq}1$ such that either an $a^n{\neq}0$ and $a^nRb=0$ or $b^n{\neq}0$ and $aRb^n=0$. In this paper, many properties of weakly semicommutative rings are introduced, some known results are extended. Especially, we show that a ring R is a strongly regular ring if and only if R is a left SF-ring and weakly semicommutative ring.

• Kyungpook Mathematical Journal
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• 제51권3호
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• pp.283-291
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• 2011

In this paper, a generalization of the class of semicommutative rings is investigated. A ring R is called central semicommutative if for any a, b ${\in}$ R, ab = 0 implies arb is a central element of R for each r ${\in}$ R. We prove that some results on semicommutative rings can be extended to central semicommutative rings for this general settings.

• 대한수학회논문집
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• 제23권3호
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• pp.333-342
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• 2008

We introduce weak Armendariz ideals which are a generalization of ideals have the weakly insertion of factors property (or simply weakly IFP) and investigate their properties. Moreover, we prove that, if I is a weak Armendariz ideal of R, then I[x] is a weak Armendariz ideal of R[x]. As a consequence, we show that, R is weak Armendariz if and only if R[x] is a weak Armendariz ring. Also we obtain a generalization of [8] and [9].

• 대한수학회논문집
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• 제35권2호
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• pp.447-454
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• 2020

The studies of reversible and NI rings have done important roles in noncommutative ring theory. A ring R shall be called QRUR if ab = 0 for a, b ∈ R implies that ba is contained in the upper nilradical of R, which is a generalization of the NI ring property. In this article we investigate the structure of QRUR rings and examine the QRUR property of several kinds of ring extensions including matrix rings and polynomial rings. We also show that if there exists a weakly semicommutative ring but not QRUR, then Köthe's conjecture does not hold.

• Korean Journal of Mathematics
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• 제22권2호
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• pp.279-288
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• 2014

The studies of reversible and 2-primal rings have done important roles in noncommutative ring theory. We in this note introduce the concept of quasi-reversible-over-prime-radical (simply, QRPR) as a generalization of the 2-primal ring property. A ring is called QRPR if ab = 0 for $a,b{\in}R$ implies that ab is contained in the prime radical. In this note we study the structure of QRPR rings and examine the QRPR property of several kinds of ring extensions which have roles in noncommutative ring theory.