• 제목, 요약, 키워드: multiplication module

### Multiplication Modules and characteristic submodules

• Park, Young-Soo;Chol, Chang-Woo
• 대한수학회보
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• v.32 no.2
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• pp.321-328
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• 1995
• In this note all are commutative rings with identity and all modules are unital. Let R be a ring. An R-module M is called a multiplication module if for every submodule N of M there esists an ideal I of R such that N = IM. Clearly the ring R is a multiplication module as a module over itself. Also, it is well known that invertible and more generally profective ideals of R are multiplication R-modules (see [11, Theorem 1]).

### SOME PROPERTIES OF GR-MULTIPLICATION MODULES

• Park, Seungkook
• Korean Journal of Mathematics
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• v.20 no.3
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• pp.315-321
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• 2012
• In this paper, we provide the necessary and sufficient conditions for a faithful graded module to be a graded multiplication module and for a graded submodule of a faithful gr-multiplication to be gr-essential.

### ON MULTIPLICATION MODULES (II)

• Cho, Yong-Hwan
• 대한수학회논문집
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• v.13 no.4
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• pp.727-733
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• 1998
• In this short paper we shall find some properties on multiplication modules and prove three theorems.

### Direct sum decompositions of indecomposable injective modules

• Lee, Sang-Cheol
• 대한수학회보
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• v.35 no.1
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• pp.33-43
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• 1998
• Matlis posed the following question in 1958: if N is a direct summand of a direct sum M of indecomposable injectives, then is N itself a direct sum of indecomposable innjectives\ulcorner It will be proved that the Matlis problem has an affirmative answer when M is a multiplication module, and that a weaker condition then that of M being a multiplication module can be given to module M when M is a countable direct sum of indecomposable injectives.

### A GENERALIZATION OF MULTIPLICATION MODULES

• Perez, Jaime Castro;Montes, Jose Rios;Sanchez, Gustavo Tapia
• 대한수학회보
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• v.56 no.1
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• pp.83-102
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• 2019
• For $M{\in}R-Mod$, $N{\subseteq}M$ and $L{\in}{\sigma}[M]$ we consider the product $N_ML={\sum}_{f{\in}Hom_R(M,L)}\;f(N)$. A module $N{\in}{\sigma}[M]$ is called an M-multiplication module if for every submodule L of N, there exists a submodule I of M such that $L=I_MN$. We extend some important results given for multiplication modules to M-multiplication modules. As applications we obtain some new results when M is a semiprime Goldie module. In particular we prove that M is a semiprime Goldie module with an essential socle and $N{\in}{\sigma}[M]$ is an M-multiplication module, then N is cyclic, distributive and semisimple module. To prove these results we have had to develop new methods.

### THE PRODUCT OF MULTIPLICATION SUBMODULES

• 호남수학학술지
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• v.27 no.1
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• pp.1-8
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• 2005
• Let R be a commutative ring with non-zero identity. This paper is devoted to the study some of properties of the product of submodules of a multiplication module. Suppose N is a submodule of a multiplication R-module M. We give a condition which allows us to determine whether N is finitely generated when we assume some power of N is finitely generated.

### A REMARK ON MULTIPLICATION MODULES

• Choi, Chang-Woo;Kim, Eun-Sup
• 대한수학회보
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• v.31 no.2
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• pp.163-165
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• 1994
• Modules which satisfy the converse of Schur's lemma have been studied by many authors. In [6], R. Ware proved that a projective module P over a semiprime ring R is irreducible if and only if En $d_{R}$(P) is a division ring. Also, Y. Hirano and J.K. Park proved that a torsionless module M over a semiprime ring R is irreducible if and only if En $d_{R}$(M) is a division ring. In case R is a commutative ring, we obtain the following: An R-module M is irreducible if and only if En $d_{R}$(M) is a division ring and M is a multiplication R-module. Throughout this paper, R is commutative ring with identity and all modules are unital left R-modules. Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for each submodule N of M, there exists and ideal I of R such that N=IM. Cyclic R-modules are multiplication modules. In particular, irreducible R-modules are multiplication modules.dules.

### IDEALS AND SUBMODULES OF MULTIPLICATION MODULES

• LEE, SANG CHEOL;KIM, SUNAH;CHUNG, SANG-CHO
• 대한수학회지
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• v.42 no.5
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• pp.933-948
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• 2005

### Some Remarks on Faithful Multiplication Modules

• Lee, Dong-Soo;Lee, Hyun-Bok
• 충청수학회지
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• v.6 no.1
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• pp.131-137
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• 1993
• Let R he a commutative ring with identity and let M be a nonzero multiplication R-module. In this note we prove that M is finitely generated if M is a faithful multiplication R-module.