• 호남수학학술지
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• 제42권1호
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• pp.93-103
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• 2020

This article concerns a property of Abelain π-regular rings. A ring R shall be called right quasi-DR if for every a ∈ R there exists n ≥ 1 such that C(R)aⁿ ⊆ aR, where C(R) means the monoid of regular elements in R. The relations between the right quasi-DR property and near ring theoretic properties are investigated. We next show that the class of right quasi-DR rings is quite large.

• 대한수학회지
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• 제54권1호
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• pp.177-191
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• 2017

In this note we consider the right unit-duo ring property on the powers of elements, and introduce the concept of weakly right unit-duo ring. We investigate first the properties of weakly right unit-duo rings which are useful to the study of related studies. We observe next various kinds of relations and examples of weakly right unit-duo rings which do roles in ring theory.

• 대한수학회보
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• 제37권2호
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• pp.217-227
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• 2000

This paper is motivated by the results in [2], [10], [13] and [19]. We study some properties of generalizations of commutative rings and relations between them. We also show that for a right quasi-duo right weakly ${\pi}-regular$ ring R, R is an (S,2)-ring if and only if every idempotent in R is a sum of two units in R, which gives a generalization of [2, Theorem 4] on right quasi-duo rings. Moreover we find a condition which is equivalent to the strongly ${\pi}-regularity$ of an abelian right quasi-duo ring.

• 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.

• 충청수학회지
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• 제7권1호
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• pp.1-11
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• 1994

The theory of inverse semigroups has many features in common with the theory of groups. Many different properties of semigroup become the same condition on ring. In this paper, we want to find the properties of semigroups which is preserved by the properties of ring. Also we find that many different properties become the equivalent conditions.

• 대한수학회지
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• 제50권5호
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• pp.1083-1103
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• 2013

The concepts of reversible, right duo, and Armendariz rings are known to play important roles in ring theory and they are independent of one another. In this note we focus on a concept that can unify them, calling it a right Armendarizlike ring in the process. We first find a simple way to construct a right Armendarizlike ring but not Armendariz (reversible, or right duo). We show the difference between right Armendarizlike rings and strongly right McCoy rings by examining the structure of right annihilators. For a regular ring R, it is proved that R is right Armendarizlike if and only if R is strongly right McCoy if and only if R is Abelian (entailing that right Armendarizlike, Armendariz, reversible, right duo, and IFP properties are equivalent for regular rings). It is shown that a ring R is right Armendarizlike, if and only if so is the polynomial ring over R, if and only if so is the classical right quotient ring (if any). In the process necessary (counter)examples are found or constructed.

• 대한수학회지
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• 제42권3호
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• pp.457-470
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• 2005

Yu showed that every right (left) primitive factor ring of weakly right (left) duo rings is a division ring. It is not difficult to show that each weakly right (left) duo ring is abelian and has the classical right (left) quotient ring. In this note we first provide a left duo ring (but not weakly right duo) in spite of it being left Noetherian and local. Thus we observe conditions under which weakly one-sided duo rings may be two-sided. We prove that a weakly one-sided duo ring R is weakly duo under each of the following conditions: (1) R is semilocal with nil Jacobson radical; (2) R is locally finite. Based on the preceding case (1) we study a kind of composition length of a right or left Artinian weakly duo ring R, obtaining that i(R) is finite and $\alpha^{i(R)}R\;=\;R\alpha^{i(R)\;=\;R\alpha^{i(R)}R\;for\;all\;\alpha\;{\in}\;R$, where i(R) is the index (of nilpotency) of R. Note that one-sided Artinian rings and locally finite rings are strongly $\pi-regular$. Thus we also observe connections between strongly $\pi-regular$ weakly right duo rings and related rings, constructing available examples.

• Korean Journal of Mathematics
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• 제21권4호
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• pp.463-471
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• 2013

Let R be a ring with identity 1, $I(R){\neq}\{0\}$ be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, $f{\in}I(R)$ ($e{\neq}f$), $e+f{\in}I(R)$. In this paper, the following are shown: (1) I(R) is a finite additive set if and only if $M(R){\backslash}\{0\}$ is a complete set of primitive central idempotents, char(R) = 2 and every nonzero idempotent of R can be expressed as a sum of orthogonal primitive idempotents of R; (2) for a regular ring R such that I(R) is a finite additive set, if the multiplicative group of all units of R is abelian (resp. cyclic), then R is a commutative ring (resp. R is a finite direct product of finite field).

• East Asian mathematical journal
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• 제36권3호
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• pp.337-347
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• 2020

Let R be a ring with unity, and let I be an ideal of R. Then R is called I-semiregular if for every a ∈ R there exists b ∈ R such that ab is an idempotent of R and a - aba ∈ I. In this paper, basic properties of I-semiregularity are investigated, and some equivalent conditions to the primitivity of e are observed for an idempotent e of an I-semiregular ring R such that I∩eR = (0). For an abelian regular ring R with the ascending chain condition on annihilators of idempotents of R, it is shown that R is isomorphic to a direct product of a finite number of division rings, as a consequence of the observations.

• Kyungpook Mathematical Journal
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• 제51권1호
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• pp.93-108
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• 2011

A ring R is called left weakly np- injective if for each non-nilpotent element a of R, there exists a positive integer n such that any left R- homomorphism from $Ra^n$ to R is right multiplication by an element of R. In this paper various properties of these rings are first developed, many extending known results such as every left or right module over a left weakly np- injective ring is divisible; R is left seft-injective if and only if R is left weakly np-injective and $_RR$ is weakly injective; R is strongly regular if and only if R is abelian left pp and left weakly np- injective. We next introduce the concepts of left weakly pp rings and left weakly C2 rings. In terms of these rings, we give some characterizations of (von Neumann) regular rings such as R is regular if and only if R is n- regular, left weakly pp and left weakly C2. Finally, the relations among left C2 rings, left weakly C2 rings and left GC2 rings are given.