• Kyungpook Mathematical Journal
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• 제48권1호
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• pp.143-154
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• 2008

Lomp [9] has studied finitely generated projective modules over semilocal rings. He obtained the following: finitely generated projective modules over semilocal rings are semilocal. We shall give necessary and sufficient conditions for finitely generated modules to be semilocal modules. By using a lifting property, we also give characterizations of right perfect (semiperfect) rings. Our main results can be summarized as follows: (1) Let M be a finitely generated module. Then M has finite hollow dimension if and only if M is weakly supplemented if and only if M is semilocal. (2) A ring R is right perfect if and only if every flat right R-module is lifting and every right R-module has a flat cover if and only if every quasi-projective right R-module is lifting. (3) A ring R is semiperfect if and only if every finitely generated flat right R-module is lifting if and only if RR satisfies the lifting property for simple factor modules.

• 대한수학회보
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• 제48권5호
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• pp.1033-1039
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• 2011

Let R be a ring with identity 1, I(R) be the set of all idempotents in R and G be the group of all units of R. In this paper, we show that for any semiperfect ring R in which 2 = 1+1 is a unit, I(R) is closed under multiplication if and only if R is a direct sum of local rings if and only if the set of all minimal idempotents in R is closed under multiplication and eGe is contained in the group of units of eRe. In particular, for a left Artinian ring in which 2 is a unit, R is a direct sum of local rings if and only if the set of all minimal idempotents in R is closed under multiplication.

• 대한수학회논문집
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• 제9권3호
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• pp.513-519
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• 1994

Endomorphism rings of Aritinian modules need not be semiperfect by a result of Camps and Mena [4], which answers in the negative to a question of Crawley and Jonsson[5]. However it was shown by Camps and Dicks[3] that endomorphism rings of a certain class of Artinian modules, we investigate some interesting structures of endormorphism rings of CS-modules.

• 대한수학회지
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• 제45권2호
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• pp.425-433
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• 2008

Let R be a ring with identity $1_R$ and let U(R) denote the group of all units of R. A ring R is called locally finite if every finite subset in it generates a finite semi group multiplicatively. In this paper, some results are obtained as follows: (1) for any semilocal (hence semiperfect) ring R, U(R) is a finite (resp. locally finite) group if and only if R is a finite (resp. locally finite) ring; U(R) is a locally finite group if and only if U$(M_n(R))$ is a locally finite group where $M_n(R)$ is the full matrix ring of $n{\times}n$ matrices over R for any positive integer n; in addition, if $2=1_R+1_R$ is a unit in R, then U(R) is an abelian group if and only if R is a commutative ring; (2) for any semiperfect ring R, if E(R), the set of all idempotents in R, is commuting, then $R/J\cong\oplus_{i=1}^mD_i$ where each $D_i$ is a division ring for some positive integer m and |E(R)|=$2^m$; in addition, if 2=$1_R+1_R$ is a unit in R, then every idempotent is central.

• Kyungpook Mathematical Journal
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• 제62권3호
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• pp.407-423
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• 2022

In this paper, we determine the structure of rings that have secondary representation (called representable rings) and give some characterizations of these rings. Also, we characterize Artinian rings in terms of representable rings. Then we introduce completely representable modules, modules every non-zero submodule of which have secondary representation, and give some necessary and sufficient conditions for a module to be completely representable. Finally, we define strongly representable modules and give some conditions under which representable module is strongly representable.

• 대한수학회보
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• 제60권6호
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• pp.1463-1475
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• 2023

In this paper, we introduce a class of rings generalizing unit regular rings and being a subclass of semipotent rings, which is called idempotent unit regular. We call a ring R an idempotent unit regular ring if for all r ∈ R - J(R), there exist a non-zero idempotent e and a unit element u in R such that er = eu, where this condition is left and right symmetric. Thus, we have also that there exist a non-zero idempotent e and a unit u such that ere = eue for all r ∈ R - J(R). Various basic characterizations and properties of this class of rings are proved and it is given the relationships between this class of rings and some well-known classes of rings such as semiperfect, clean, exchange and semipotent. Moreover, we obtain some results about when the endomorphism ring of a module in a class of left R-modules X is idempotent unit regular.

• East Asian mathematical journal
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• 제34권5호
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• pp.615-621
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• 2018

Let R be a ring with the unity 1, and let e be an idempotent of R. In this paper, we discuss some symmetric property for the set $\{(a_1,a_2,{\cdots},a_n){\in}R^n:a_1a_2{\cdots}a_n=e\}$. We here investigate some properties of those rings with such a symmetric property for an arbitrary idempotent e; some of our results turn out to generalize some known results observed already when n = 2 and e = 0, 1 by several authors. We also focus especially on the case when n = 3 and e = 1. As consequences of our observation, we also give some equivalent conditions to the commutativity for some classes of rings, in terms of the symmetric property.

• 대한수학회보
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• 제56권1호
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• pp.179-186
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• 2019

This article mainly concentrates on the open question whether a right self-injective ring R is necessary QF if $R/S_l$ is left Goldie. It is answered affirmatively under the condition $S_l{\subseteq}S_r$, where $S_l$ and $S_r$ denote the left socle and right socle of R respectively. And the original condition "right self-injective" can be weakened to "right CS and right P-injective". It is also proved that a semiperfect, left and right mininjective ring R is QF if $S_r{\subseteq}^{ess}$ $R_R$ and $R/S_l$ is left Goldie.

• Kyungpook Mathematical Journal
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• 제55권2호
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• pp.289-300
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• 2015

In [9], the author extends the definition of lifting and supplemented modules to ${\delta}$-lifting and ${\delta}$-supplemented by replacing "small submodule" with "${\delta}$-small submodule" introduced by Zhou in [13]. The aim of this paper is to show new properties of ${\delta}$-lifting and ${\delta}$-supplemented modules. Especially, we show that any finite direct sum of ${\delta}$-hollow modules is ${\delta}$-supplemented. On the other hand, the notion of amply ${\delta}$-supplemented modules is studied as a generalization of amply supplemented modules and several properties of these modules are given. We also prove that a module M is Artinian if and only if M is amply ${\delta}$-supplemented and satisfies Descending Chain Condition (DCC) on ${\delta}$-supplemented modules and on ${\delta}$-small submodules. Finally, we obtain the following result: a ring R is right Artinian if and only if R is a ${\delta}$-semiperfect ring which satisfies DCC on ${\delta}$-small right ideals of R.