• Journal of the Chungcheong Mathematical Society
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• v.32 no.4
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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.

• Korean Journal of Mathematics
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• v.28 no.1
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• pp.65-73
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• 2020

This article concerns a property of local rings and domains. A ring R is called weakly local if for every a ∈ R, a is regular or 1-a is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.

• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.447-459
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• 2016

Let D be an integral domain, {$X_{\alpha}$} be a nonempty set of indeterminates over D, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}_1}$ be the first type power series ring over D. We show that if D is a t-SFT $Pr{\ddot{u}}fer$ v-multiplication domain, then $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_{1_{D-\{0\}}}$ is a Krull domain, and $D{\mathbb{[}}\{X_{\alpha}\}{\mathbb{]}}_1$ is a $Pr{\ddot{u}}fer$ v-multiplication domain if and only if D is a Krull domain.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.387-396
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• 2016

An $n{\times}n$ matrix A over a commutative ring is strongly clean provided that it can be written as the sum of an idempotent matrix and an invertible matrix that commute. Let R be an arbitrary commutative ring, and let $A(x){\in}M_n(R[[x]])$. We prove, in this note, that $A(x){\in}M_n(R[[x]])$ is strongly clean if and only if $A(0){\in}M_n(R)$ is strongly clean. Strongly clean matrices over quotient rings of power series are also determined.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1613-1615
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• 2016

It is proved that $k[X_1,{\ldots},X_v ]$ localized at the ideal ($X_1,{\ldots},X_v$ ), where k is a field and $X_1,{\ldots},X_v$ indeterminates, is not weakly quasi-complete for $v{\geq}2$, thus proving a conjecture of D. D. Anderson and solving a problem from "Open Problems in Commutative Ring Theory" by Cahen, Fontana, Frisch, and Glaz.

• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.663-683
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• 2015

In this note we study the structures of power-serieswise Armendariz rings and IFP rings when they are skewed by ring endomor-phisms (or automorphisms). We call such rings skew power-serieswise Armendariz rings and skew IFP rings, respectively. We also investigate relationships among them and construct necessary examples in the process. The results argued in this note can be extended to the ordinary ring theoretic properties of power-serieswise Armendariz rings, IFP rings, and near-related rings.

• Bulletin of the Korean Mathematical Society
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• v.38 no.4
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• pp.623-633
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• 2001

In this note we study the prime radicals of formal power series rings, and the shapes of them under the condition that the prime radical is nilpotent. Furthermore we observe the condition structurally, adding related examples to the situations that occur naturally in the process.

• Communications of the Korean Mathematical Society
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• v.23 no.2
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• pp.161-167
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• 2008

A ring R is called an $I_0$-ring if each left ideal not contained in the Jacobson radical J(R) contains a non-zero idempotent. If, in addition, idempotents can be lifted modulo J(R), R is called an I-ring or a potent ring. We study whether these properties are inherited by some related rings. Also, we investigate the structure of potent rings.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.1067-1079
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• 2012

Let M be a right R-module, (S, ${\leq}$) a strictly totally ordered monoid which is also artinian and ${\omega}:S{\rightarrow}Aut(R)$ a monoid homomorphism, and let $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ denote the generalized inverse polynomial module over the skew generalized power series ring [[$R^{S,{\leq}},{\omega}$]]. In this paper, we prove that $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same uniform dimension as its coefficient module $M_R$, and that if, in addition, R is a right perfect ring and S is a chain monoid, then $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same couniform dimension as its coefficient module $M_R$.

• The Mathematical Education
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• v.13 no.1
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• pp.41-42
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• 1974