• Journal of the Korean Mathematical Society
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• 제60권3호
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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.

• Communications of the Korean Mathematical Society
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• 제11권3호
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• pp.569-573
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• 1996

We study some special cases of Chow groups of a ramified complete regular local ring R of dimension n. We prove that (a) for codimension 3 Gorenstein ideal I, [I] = 0 in $A_{n-3}(R)$ and (b) for a particular class of almost complete intersection prime ideals P of height i, [P] = 0 in $A_{n-i}(R)$.

• Honam Mathematical Journal
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• 제30권4호
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• pp.617-629
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• 2008

In this paper, the authors have found an equivalent condition of the endomorphism ring End(M) of a projective module M being von Neumann regular(Theorem 1.14) and found an equivalent condition of any associative ring R being von Neumann regular (Theorem 1.13).

• Bulletin of the Korean Mathematical Society
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• 제29권1호
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• pp.137-143
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• 1992

The purpose of this paper is to provide the affirmative solution of the following conjecture due to Davis and Geramita. Conjecture; Let A=R[T] be a polynomial ring in one variable, where R is a regular local ring of dimension d. Then maximal ideals in A are complete intersection. Geramita has proved that the conjecture is true when R is a regular local ring of dimension 2. Whatwadekar has rpoved that conjecture is true when R is a formal power series ring over a field and also when R is a localization of an affine algebra over an infinite perfect field. Nashier also proved that conjecture is true when R is a local ring of D[ $X_{1}$,.., $X_{d-1}$] at the maximal ideal (.pi., $X_{1}$,.., $X_{d-1}$) where (D,(.pi.)) is a discrete valuation ring with infinite residue field. The methods to establish our results are following from Nashier's method. We divide this paper into three sections. In section 1 we state Theorems without proofs which are used in section 2 and 3. In section 2 we prove some lemmas and propositions which are used in proving our results. In section 3 we prove our main theorem.eorem.rem.

• Bulletin of the Korean Mathematical Society
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• 제55권1호
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• pp.25-40
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• 2018

We focus on the structure of the set of noncentral idempotents whose role is similar to one of central idempotents. We introduce the concept of quasi-Abelian rings which unit-regular rings satisfy. We first observe that the class of quasi-Abelian rings is seated between Abelian and direct finiteness. It is proved that a regular ring is directly finite if and only if it is quasi-Abelian. It is also shown that quasi-Abelian property is not left-right symmetric, but left-right symmetric when a given ring has an involution. Quasi-Abelian property is shown to do not pass to polynomial rings, comparing with Abelian property passing to polynomial rings.

• Bulletin of the Korean Mathematical Society
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• 제32권2호
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• pp.145-152
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• 1995

From now on, we assume that a ring R has an identity 1. We have the following Lemma from Park[2].

• Journal of the Chungcheong Mathematical Society
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• 제19권3호
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• pp.225-229
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• 2006

In this paper, we introduce some concepts of ${\Gamma}$-near-ring and obtain their properties on ${\Gamma}$-near-rings through regularity conditions.

• Journal of the Korean Mathematical Society
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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.

• Journal of the Chungcheong Mathematical Society
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• 제22권3호
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• pp.293-297
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• 2009

We will introduce some properties of strongly reduced near-rings and the notion of left $\pi$-regular near-ring. Also, we will study for right $\pi$-regular, strongly left $\pi$-regular, strongly right $\pi$-regular and strongly $\pi$- regular. Next, we may characterize the strongly $\pi$-regular near-rings with related strong reducibility.

• Journal of the Korean Mathematical Society
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• 제51권4호
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• pp.751-771
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• 2014

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring properties pass to power series rings and polynomial rings. It is also shown that ${\pi}$-regular rings are strongly ${\pi}$-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommutative rings appropriate for the situations occurred naturally in studying standard ring theoretic properties.