• Journal of applied mathematics & informatics
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• 제42권1호
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• pp.199-206
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• 2024

The goal of this study is to bring out the following conclusion: Let 𝓡 be a non-commutative prime ring of characteristic not two and 𝓓 be a bi-semiderivation on 𝓡 with a function 𝖋 (surjective). If 𝓓 acts as an endomorphism or as an anti-endomorphism, then 𝓓 = 0 on 𝓡.

• 대한수학회논문집
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• 제11권2호
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• pp.311-314
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• 1996

Let G be an abelian group of finite rink and E be the endomorphism ring of G. Then G is a left E-module by defining $f\cdota = f(a)$ for $f \in E$ and $a \in G$. In this case a condition for an E-module G to be regular is given.

• East Asian mathematical journal
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• 제6권2호
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• pp.167-182
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• 1990

The relationships between submodules of a module and ideals of the endomorphism ring of a module had been studied in [1]. For a submodule L of a moudle M, the set $I^L$ of all endomorphisms whose images are contained in L is a left ideal of the endomorphism ring End (M) and for a submodule N of M, the set $I_N$ of all endomorphisms whose kernels contain N is a right ideal of End (M). In this paper, author defines an H-invariant module and proves that every submodule of an H-invariant module is the image and kernel of unique endomorphisms. Every ideal $I^L(I_N)$ of the endomorphism ring End(M) when M is H-invariant is a left (respectively, right) principal ideal of End(M). From the above results, if a module M is H-invariant then each left, right, or both sided ideal I of End(M) is an intersection of a left, right, or both sided principal ideal and I itself appropriately. If M is an H-invariant module then the ACC on the set of all left ideals of type $I^L$ implies the ACC on M. Also if the set of all right ideals of type $I^L$ has DCC, then H-invariant module M satisfies ACC. If the set of all left ideals of type $I^L$ satisfies DCC, then H-invariant module M satisfies DCC. If the set of all right ideals of type $I_N$ satisfies ACC then H-invariant module M satisfies DCC. Therefore for an H-invariant module M, if the endomorphism ring End(M) is left Noetherian, then M satisfies ACC. And if End(M) is right Noetherian then M satisfies DCC. For an H-invariant module M, if End(M) is left Artinian then M satisfies DCC. Also if End(M) is right Artinian then M satisfies ACC.

• 대한수학회보
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• 제45권2호
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• pp.285-297
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• 2008

For an endomorphism ${\alpha}$ of a ring R, the endomorphism ${\alpha}$ is called semicommutative if ab=0 implies $aR{\alpha}(b)$=0 for a ${\in}$ R. A ring R is called ${\alpha}$-semicommutative if there exists a semicommutative endomorphism ${\alpha}$ of R. In this paper, various results of semicommutative rings are extended to ${\alpha}$-semicommutative rings. In addition, we introduce the notion of an ${\alpha}$-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring $R[[x;\;{\alpha}]]$. We show that a number of interesting properties of a ring R transfer to its the skew power series ring $R[[x;\;{\alpha}]]$ and vice-versa such as the Baer property and the p.p.-property, when R is ${\alpha}$-skew power series Armendariz. Several known results relating to ${\alpha}$-rigid rings can be obtained as corollaries of our results.

• 대한수학회보
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• 제38권1호
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• pp.149-156
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• 2001

In this paper, we initiate the investigation of ring in which all the additive endomorphisms are generated by ring endomorphisms (AGE-rings). This study was motivated by the work on the Sullivan’s Research Problem [11]: Characterize those rings in which every additive endomorphism is a ring endomorphism (AE-rings). The purpose of this paper is to obtain a certain characterization of AGE-rings, and investigate some relations between AGE and LSD-generated rings.

• 대한수학회보
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• 제54권1호
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• pp.319-329
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• 2017

Let I denote an ideal of a Noetherian local ring (R, m). Let M denote a finitely generated R-module. We study the endomorphism ring of the local cohomology module $H^c_I(M)$, c = grade(I, M). In particular there is a natural homomorphism $$Hom_{\hat{R}^I}({\hat{M}}^I,\;{\hat{M}}^I){\rightarrow}Hom_R(H^c_I(M),\;H^c_I(M))$$, $where{\hat{\cdot}}^I$ denotes the I-adic completion functor. We provide sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J{\subset}I$ with the property grade(I, M) = grade(J, M). Our results extends constructions known in the case of M = R (see e.g. [8], [17], [18]).

• 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.

• 대한수학회지
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• 제42권1호
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• pp.53-63
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• 2005

Let R be a ring R and ${\sigma}$ be an endomorphism of R. R is called ${\sigma}$-rigid (resp. reduced) if $a{\sigma}r(a) = 0 (resp{\cdot}a^2 = 0)$ for any $a{\in}R$ implies a = 0. An ideal I of R is called a ${\sigma}$-ideal if ${\sigma}(I){\subseteq}I$. R is called ${\sigma}$-quasi-Baer (resp. right (or left) ${\sigma}$-p.q.-Baer) if the right annihilator of every ${\sigma}$-ideal (resp. right (or left) principal ${\sigma}$-ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[$x;{\sigma}$] of a ring R is investigated as follows: For a ${\sigma}$-rigid ring R, (1) R is ${\sigma}$-quasi-Baer if and only if A is quasi-Baer if and only if A is $\={\sigma}$-quasi-Baer for every extended endomorphism $\={\sigma}$ on A of ${\sigma}$ (2) R is right ${\sigma}$-p.q.-Baer if and only if R is ${\sigma}$-p.q.-Baer if and only if A is right p.q.-Baer if and only if A is p.q.-Baer if and only if A is $\={\sigma}$-p.q.-Baer if and only if A is right $\={\sigma}$-p.q.-Baer for every extended endomorphism $\={\sigma}$ on A of ${\sigma}$.

• 호남수학학술지
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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).

• 대한수학회보
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• 제51권4호
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• pp.1127-1133
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• 2014

Let S be a nonempty subset of a ring R. A map $f:R{\rightarrow}R$ is called strong commutativity preserving on S if [f(x), f(y)] = [x, y] for all $x,y{\in}S$, where the symbol [x, y] denotes xy - yx. Bell and Daif proved that if a derivation D of a semiprime ring R is strong commutativity preserving on a nonzero right ideal ${\rho}$ of R, then ${\rho}{\subseteq}Z$, the center of R. Also they proved that if an endomorphism T of a semiprime ring R is strong commutativity preserving on a nonzero two-sided ideal I of R and not identity on the ideal $I{\cup}T^{-1}(I)$, then R contains a nonzero central ideal. This short note shows that the conclusions of Bell and Daif are also true without the additivity of the derivation D and the endomorphism T.