• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.813-822
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• 2014

A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.555-565
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• 2014

An ideal I of a ring R is strongly ${\pi}$-regular if for any $x{\in}I$ there exist $n{\in}\mathbb{N}$ and $y{\in}I$ such that $x^n=x^{n+1}y$. We prove that every strongly ${\pi}$-regular ideal of a ring is a B-ideal. An ideal I is periodic provided that for any $x{\in}I$ there exist two distinct m, $n{\in}\mathbb{N}$ such that $x^m=x^n$. Furthermore, we prove that an ideal I of a ring R is periodic if and only if I is strongly ${\pi}$-regular and for any $u{\in}U(I)$, $u^{-1}{\in}\mathbb{Z}[u]$.

• The Pure and Applied Mathematics
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• v.14 no.3
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• pp.153-159
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• 2007

The notion of bi-ideals in near-rings was effectively used to characterize the near-fields. Using this notion, various generalizations of regularity conditions have been studied. In this paper, we generalize further the notion of bi-ideals and introduce the notion of weak bi-ideals in near-rings and obtain various characterizations using the same in left self distributive near-rings.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.45-56
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• 2022

In this paper, we introduce and study the class of rings in which every ideal consisting entirely of zero divisors is a d-ideal, considered as a generalization of strongly duo rings. Some results including the characterization of AA-rings are given in the first section. Further, we examine the stability of these rings in localization and study the possible transfer to direct product and trivial ring extension. In addition, we define the class of d_E-ideals which allows us to characterize von Neumann regular rings.

• Kyungpook Mathematical Journal
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• v.26 no.2
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• pp.137-148
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• 1986

• East Asian mathematical journal
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• v.6 no.2
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• pp.183-192
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• 1990

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1887-1903
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• 2013

We continue the study of McCoy condition to analyze zero-dividing polynomials for the constant annihilators in the ideals generated by the coefficients. In the process we introduce the concept of ideal-${\pi}$-McCoy rings, extending known results related to McCoy condition. It is shown that the class of ideal-${\pi}$-McCoy rings contains both strongly McCoy rings whose non-regular polynomials are nilpotent and 2-primal rings. We also investigate relations between the ideal-${\pi}$-McCoy property and other standard ring theoretic properties. Moreover we extend the class of ideal-${\pi}$-McCoy rings by examining various sorts of ordinary ring extensions.

• Bulletin of the Korean Mathematical Society
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• v.37 no.2
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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.

• Honam Mathematical Journal
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• v.42 no.2
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• pp.219-234
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• 2020

The fundamental relation on a fuzzy hyperring is defined as the smallest equivalence relation, such that the quotient would be the ring, that is not commutative necessarily. In this paper, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings, where the ring is commutative with respect to both sum and product. With considering this relation on fuzzy hyperring, the set of the quotient is a commutative ring. Also, we introduce fundamental functor between the category of fuzzy hyperrings and category of commutative rings and some related properties. Eventually, we introduce α-part in fuzzy hyperring and determine some necessary and sufficient conditions so that the relation α is transitive.

• Journal of applied mathematics & informatics
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• v.41 no.6
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• pp.1173-1180
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• 2023

In this paper, rings in which essential maximal right ideals are GP-injective are studied. Whether the rings with this condition satisfy von Neumann regularity is the goal of this study. The obtained research results are twofold: First, it was shown that this regularity holds even when the reduced ring is replaced with π-IFP and NI-ring. Second, it was shown that this regularity also holds even when the maximal right ideal is changed to GW-ideal. This can be interpreted as an extension of the existing results.