• 대한수학회보
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• 제47권4호
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• pp.715-726
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• 2010

We continue the study of fully idempotent rings initiated by Courter. It is shown that a (semi)prime ring, but not fully idempotent, can be always constructed from any (semi)prime ring. It is shown that the full idempotence is both Morita invariant and a hereditary radical property, obtaining $hs(Mat_n(R))\;=\;Mat_n(hs(R))$ for any ring R where hs(-) means the sum of all fully idempotent ideals. A non-semiprimitive fully idempotent ring with identity is constructed from the Smoktunowicz's simple nil ring. It is proved that the full idempotence is preserved by the classical quotient rings. More properties of fully idempotent rings are examined and necessary examples are found or constructed in the process.

• 대한수학회보
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• 제61권2호
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• pp.291-299
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• 2024

We study some factorization properties of the idealization R(+)M of a module M in a commutative ring R which is not necessarily a domain. We show that R(+)M is ACCP if and only if R is ACCP and M satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which R(+)M is a BFR. We also characterize the idealization rings which are UFRs.

• 대한수학회보
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• 제55권4호
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• pp.1179-1187
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• 2018

We study the class of rings R with the property that for $x{\in}R$ at least one of the elements x and 1 + x are tripotent. We prove that a commutative ring has this property if and only if it is a subring of a direct product $R_0{\times}R_1{\times}R_2$ such that $R_0/J(R_0){\cong}{\mathbb{z}}_2$, for every $x{\in}J(R_0)$ we have $x^2=2x$, $R_1$ is a Boolean ring, and $R_3$ is a subring of a direct product of copies of ${\mathbb{z}}_3$.

• 대한수학회보
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• 제31권2호
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• pp.210-214
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• 1994

The purpose of this paper is to investigate the relationship between the depths of the Rees algebra R[It] and the associated graded ring g $r_{I}$(R) of an ideal I in a local ring (R,m) of dim(R) > 0. The relationship between the Cohen-Macaulayness of these two rings has been studied extensively. Let (R, m) be a local ring and I an ideal of R. An ideal J contained in I is called a reduction of I if J $I^{n}$ = $I^{n+1}$ for some integer n.geq.0. A reduction J of I is called a minimal reduction of I. The reduction number of I with respect to J is defined by (Fig.) S. Goto and Y.Shimoda characterized the Cohen-Macaulay property of the Rees algebra of the maximal ideal of a Cohen-Macaulay local ring in terms of the Cohen-Macaulay property of the associated graded ring of the maximal ideal and the reduction number of that maximal ideal. Let us state their theorem.m.m.

• 대한수학회논문집
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• 제36권1호
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• pp.27-39
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• 2021

We discuss the condition that every nonzero right annihilator of an element contains a nonzero ideal, as a generalization of the insertion-of-factors-property. A ring with such condition is called right AP. We prove that a ring R is right AP if and only if Dn(R) is right AP for every n ≥ 2, where Dn(R) is the ring of n by n upper triangular matrices over R whose diagonals are equal. Properties of right AP rings are investigated in relation to nilradicals, prime factor rings and minimal order.

• 호남수학학술지
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• 제3권1호
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• pp.19-21
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• 1981

In this note, with respect to the quotient ring, we give a property of those rings in which every prime ideal is maximal.

• 대한수학회보
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• 제59권3호
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• pp.529-545
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• 2022

This article concerns a ring property that arises from combining one-sided duo factor rings and centers. A ring R is called right CIFD if R/I is right duo by some proper ideal I of R such that I is contained in the center of R. We first see that this property is seated between right duo and right π-duo, and not left-right symmetric. We prove, for a right CIFD ring R, that W(R) coincides with the set of all nilpotent elements of R; that R/P is a right duo domain for every minimal prime ideal P of R; that R/W(R) is strongly right bounded; and that every prime ideal of R is maximal if and only if R/W(R) is strongly regular, where W(R) is the Wedderburn radical of R. It is also proved that a ring R is commutative if and only if D₃(R) is right CIFD, where D₃(R) is the ring of 3 by 3 upper triangular matrices over R whose diagonals are equal. Furthermore, we show that the right CIFD property does not pass to polynomial rings, and that the polynomial ring over a ring R is right CIFD if and only if R/I is commutative by a proper ideal I of R contained in the center of R.

• Korean Journal of Mathematics
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• 제22권4호
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• pp.611-619
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• 2014

We in this note study a ring theoretic property which unifies Armendariz and IFP. We call this new concept INFP. We first show that idempotents and nilpotents are connected by the Abelian ring property. Next the structure of INFP rings is studied in relation to several sorts of algebraic systems.

• 대한수학회논문집
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• 제36권2호
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• pp.209-227
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• 2021

In this paper, we deal with the question that what kind of properties does a ring gain when it satisfies symmetricity or reversibility by the way of nilpotent elements? By the motivation of this question, we approach to symmetric and reversible property of rings via nilpotents. For symmetricity, we call a ring R middle right-(resp. left-)nil symmetric (mr-nil (resp. ml-nil) symmetric, for short) if abc = 0 implies acb = 0 (resp. bac = 0) for a, c ∈ R and b ∈ nil(R) where nil(R) is the set of all nilpotent elements of R. It is proved that mr-nil symmetric rings are abelian and so directly finite. We show that the class of mr-nil symmetric rings strictly lies between the classes of symmetric rings and weak right nil-symmetric rings. For reversibility, we introduce left (resp. right) N-reversible ideal I of a ring R if for any a ∈ nil(R), b ∈ R, being ab ∈ I implies ba ∈ I (resp. b ∈ nil(R), a ∈ R, being ab ∈ I implies ba ∈ I). A ring R is called left (resp. right) N-reversible if the zero ideal is left (resp. right) N-reversible. Left N-reversibility is a generalization of mr-nil symmetricity. We exactly determine the place of the class of left N-reversible rings which is placed between the classes of reversible rings and CNZ rings. We also obtain that every left N-reversible ring is nil-Armendariz. It is observed that the polynomial ring over a left N-reversible Armendariz ring is also left N-reversible.

• 충청수학회지
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• 제32권4호
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• pp.401-408
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• 2019

A ring R is called right (resp., left) nilpotent-duo if N(R)a ⊆ aN(R) (resp., aN(R) ⊆ N(R)a) for every a ∈ R, where N(R) is the set of all nilpotents in R. In this article we provide other proofs of known results and other computations for known examples in relation with right nilpotent-duo property. Furthermore we show that the left nilpotent-duo property does not go up to a kind of matrix ring.