• 대한수학회지
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• 제45권3호
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• pp.741-756
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• 2008

In this paper, we introduce a new class of rings, SB-rings. We establish various properties of this concept. These shows that, in several respects, SB-rings behave like rings satisfying unit 1-stable range. We will give necessary and sufficient conditions under which a semilocal ring is a SB-ring. Furthermore, we extend these results to exchange rings with all primitive factors artinian. For such rings, we observe that the concept of the SB-ring coincides with Goodearl-Menal condition. These also generalize the results of Huh et al., Yu and the author on rings generated by their units.

• 대한수학회논문집
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• 제34권3호
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• pp.719-727
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• 2019

This article concerns a generalization of Armendariz rings that is done by restricting the degree to one. We shall call such rings, as to satisfy this property, partial-Armendariz. We first show that partial-Armendariz rings are between Armendariz rings and weak Armendariz rings. The basic structures of partial-Armendariz rings are investigated, and the relations between partial-Armendariz rings and near related ring properties are also studied.

• 대한수학회지
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• 제58권2호
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• pp.309-326
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• 2021

In near-ring theory several different types of primitivity exist. These all imply several different types of primeness. In case of near-rings with DCCN most of the types of primeness are known to imply primitivity of a certain kind. We are able to show that also so called 1-prime near-rings imply 1-primitivity. This enables us to classify maximal ideals in near-rings with chain condition with the concept of 1-primeness which leads to further results in the structure theory of near-rings.

• 대한수학회보
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• 제59권5호
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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.

• 대한수학회지
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• 제60권2호
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• pp.327-339
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• 2023

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. In this article we introduce a new class of ring, called S-multiplication rings which are S-versions of multiplication rings. An R-module M is said to be S-multiplication if for each submodule N of M, sN ⊆ JM ⊆ N for some s ∈ S and ideal J of R (see for instance [4, Definition 1]). An ideal I of R is called S-multiplication if I is an S-multiplication R-module. A commutative ring R is called an S-multiplication ring if each ideal of R is S-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and S-P IR. Moreover, we generalize some properties of multiplication rings to S-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.

• 대한수학회논문집
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• 제32권1호
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• pp.19-27
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• 2017

We define and completely describe the structure of invo-clean rings having identity. We show that these rings are clean but not (weakly) nil-clean and thus they possess independent properties than these obtained by Diesl in [7] and by Breaz-Danchev-Zhou in [2].

• 대한수학회보
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• 제44권4호
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• pp.777-788
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• 2007

An endomorphism ${\alpha}$ of a ring R is called right(left) symmetric if whenever abc=0 for a, b, c ${\in}$ R, $ac{\alpha}(b)=0({\alpha}(b)ac=0)$. A ring R is called right(left) ${\alpha}-symmetric$ if there exists a right(left) symmetric endomorphism ${\alpha}$ of R. The notion of an ${\alpha}-symmetric$ ring is a generalization of ${\alpha}-rigid$ rings as well as an extension of symmetric rings. We study characterizations of ${\alpha}-symmetric$ rings and their related properties including extensions. The relationship between ${\alpha}-symmetric$ rings and(extended) Armendariz rings is also investigated, consequently several known results relating to ${\alpha}-rigid$ and symmetric rings can be obtained as corollaries of our results.

• 대한수학회지
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• 제56권5호
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• pp.1403-1418
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• 2019

Let R be a commutative ring with unity. If f(x) is a zero-divisor polynomial such that $f(x)=c_f f_1(x)$ with $c_f{\in}R$ and $f_1(x)$ is not zero-divisor, then $c_f$ is called an annihilating content for f(x). In this case $Ann(f)=Ann(c_f )$. We defined EM-rings to be rings with every zero-divisor polynomial having annihilating content. We showed that the class of EM-rings includes integral domains, principal ideal rings, and PP-rings, while it is included in Armendariz rings, and rings having a.c. condition. Some properties of EM-rings are studied and the zero-divisor graphs ${\Gamma}(R)$ and ${\Gamma}(R[x])$ are related if R was an EM-ring. Some properties of annihilating contents for polynomials are extended to formal power series rings.

• Korean Journal of Mathematics
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• 제31권4호
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• pp.405-417
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• 2023

The RIP of rings was introduced by Kwak and Lee as a generalization of the one-sided idempotent-reflexivity property. In this study, we focus on rings in which all one-sided semicentral idempotents are central, and we refer to them as quasi-Abelian rings, extending the concept introduced by RIP. We establish that quasi-Abelianity extends to various types of rings, including polynomial rings, power series rings, Laurent series rings, matrices, and certain subrings of triangular matrix rings. Furthermore, we provide comprehensive proofs for several results that hold for RIP and are also satisfied by the quasi-Abelian property. Additionally, we investigate the structural properties of minimal non-Abelian quasi-Abelian rings.

• 대한수학회보
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• 제48권1호
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• pp.157-167
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• 2011

For a ring endomorphism of a ring R, Krempa called $\alpha$ rigid endomorphism if $a{\alpha}(a)$ = 0 implies a = 0 for a $\in$ R, and Hong et al. called R an $\alpha$-rigid ring if there exists a rigid endomorphism $\alpha$. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., $\alpha$-Armendariz rings and $\alpha$-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between $\alpha$-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an $\alpha$-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.