• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.629-633
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• 2003

We investigate in this paper rings containing a finitely generated p-injective maximal left ideal. We show that if R is a semiprime ring containing a finitely generated p-injective maximal left ideal, then R is a left p-injective ring. Using this result we are able to give a new characterization of von Neumann regular rings with nonzero socle.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1237-1246
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• 2022

In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it a mid ring. Also, we provide new characterizations for von Neumann regular and zero-dimensional rings. Moreover, some results about mp-ring are given. Finally, a characterization for mid rings is provided. Then it is shown that the class of mid rings is strictly between the class of reduced mp-rings (p.f. rings) and the class of mp-rings.

• Journal of the Korean Mathematical Society
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• v.60 no.3
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• pp.521-536
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• 2023

Let R be a commutative ring with identity and S a multiplicative subset of R. In this paper, we introduce and study the notions of u-S-pure u-S-exact sequences and uniformly S-absolutely pure modules which extend the classical notions of pure exact sequences and absolutely pure modules. And then we characterize uniformly S-von Neumann regular rings and uniformly S-Noetherian rings using uniformly S-absolutely pure modules.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1177-1190
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• 2022

Let R be a commutative ring with identity. We call the ring R to be an almost quasi-coherent ring if for any finite set of elements α₁, …, α_p and a of R, there exists a positive integer m such that the ideals $\bigcap{_{i=1}^{p}}\;R{\alpha}^m_i$ and Ann_R(α^m) are finitely generated, and to be almost von Neumann regular rings if for any two elements a and b in R, there exists a positive integer n such that the ideal (αⁿ, bⁿ) is generated by an idempotent element. This paper establishes necessary and sufficient conditions for the Nagata's idealization and the amalgamated algebra to inherit these notions. Our results allow us to construct original examples of rings satisfying the above-mentioned properties.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.399-407
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• 2022

In this paper, we continue to study the von Neumann regularity of rings whose essential maximal right ideals are GP-injective. It is proved that the following statements are equivalent: (1) R is strongly regular; (2) R is a 2-primal ring whose essential maximal right ideals are GP-injective; (3) R is a right (or left) quasi-duo ring whose essential maximal right ideals are GP-injective. Moreover, it is shown that R is strongly regular if and only if R is a strongly right (or left) bounded ring whose essential maximal right ideals are GP-injective. Finally, we prove that a PI-ring whose essential maximal right ideals are GP-injective is strongly π-regular.

• Korean Journal of Mathematics
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• v.20 no.4
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• pp.423-431
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• 2012

There have been some study characterizing monoids by homological classification using the properties around projectivity, injectivity, or regularity of acts. In particular Kilp and Knauer([4]) have analyzed monoids over which all acts with one of the properties around projectivity or injectivity are regular. However Kilp and Knauer left over problems of characterization of monoids over which all regular right S-acts are (weakly) at, (weakly) injective or faithful. Among these open problems, Liu([3]) proved that all regular right S-acts are (weakly) at if and only if es is a von Neumann regular element of S for all $s{\in}S$ and $e^2=e{\in}T$, and that all regular right S-acts are faithful if and only if all right ideals eS, $e^2=e{\in}T$, are faithful. But it still remains an open question to characterize over which all regular right S-acts are weakly injective or injective. Hence the purpose of this study is to investigate the relations between regular right S-acts and weakly injective right S-acts, and then characterize the monoid over which all regular right S-acts are weakly injective.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.223-234
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• 2003

This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right YJ-injective, right and left V-ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that $_{A}$A/J is p-injective, then the ring A/J is strongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p.p.ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left YJ-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular : (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, $_{A}$M/Z(M) is projective. ((5) remains valid if 《continuous》 is replaced by 《self-injective》 and 《cyclic》 is replaced by 《finitely generated》. Finally, we have the following two equivalent properties for A to be von Neumann regula. : (a) A is left non-singular such that every finitely generated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change 《left non-singular》 into 《right non-singular》in (a).(a).

• Communications of the Korean Mathematical Society
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• v.17 no.3
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• pp.409-422
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• 2002

We investigate the relationship between various generalizations of von Neumann regularity condition and the condition that every prime ideal is maximal in exchange rings.

• 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.

• Journal of the Korean Mathematical Society
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• v.19 no.2
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• pp.97-104
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• 1983