• Title/Summary/Keyword: Multiplication modules

### MULTIPLICATION MODULES OVER PULLBACK RINGS (I)

• ATANI, SHAHABADDIN EBRAHIMI;LEE, SANG CHEOL
• Honam Mathematical Journal
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• v.28 no.1
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• pp.69-81
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• 2006
• First, we give a complete description of the multiplication modules over local Dedekind domains. Second, if R is the pullback ring of two local Dedekind domains over a common factor field then we give a complete description of separated multiplication modules over R.

### ON MULTIPLICATION MODULES (II)

• Cho, Yong-Hwan
• Communications of the Korean Mathematical Society
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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.

### ASSOCIATED PRIME SUBMODULES OF A MULTIPLICATION MODULE

• Lee, Sang Cheol;Song, Yeong Moo;Varmazyar, Rezvan
• Honam Mathematical Journal
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• v.39 no.2
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• pp.275-296
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• 2017
• All rings considered here are commutative rings with identity and all modules considered here are unital left modules. A submodule N of an R-module M is said to be extended to M if $N=aM$ for some ideal a of R and it is said to be fully invariant if ${\varphi}(L){\subseteq}L$ for every ${\varphi}{\in}End(M)$. An R-module M is called a [resp., fully invariant] multiplication module if every [resp., fully invariant] submodule is extended to M. The class of fully invariant multiplication modules is bigger than the class of multiplication modules. We deal with prime submodules and associated prime submodules of fully invariant multiplication modules. In particular, when M is a nonzero faithful multiplication module over a Noetherian ring, we characterize the zero-divisors of M in terms of the associated prime submodules, and we show that the set Aps(M) of associated prime submodules of M determines the set $Zdv_M(M)$ of zero-dvisors of M and the support Supp(M) of M.

### A GENERALIZATION OF MULTIPLICATION MODULES

• Perez, Jaime Castro;Montes, Jose Rios;Sanchez, Gustavo Tapia
• Bulletin of the Korean Mathematical Society
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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.

### ON MULTIPLICATION MODULES(V)

• Cho, Yong-Hwan;Kim, Youn-Hyung
• Honam Mathematical Journal
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• v.30 no.2
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• pp.363-368
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• 2008
• In this paper, we find some properties of multiplication modules and give an alternative proof of a Theorem of P.F. Smith.

### FINITELY GENERATED gr-MULTIPLICATION MODULES

• Park, Seungkook
• Journal of the Chungcheong Mathematical Society
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• v.25 no.4
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• pp.717-723
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• 2012
• In this paper, we investigate when gr-multiplication modules are finitely generated and show that if M is a finitely generated gr-multiplication R-module then there is a lattice isomorphism between the lattice of all graded ideals I of R containing ann(M) and the lattice of all graded submodules of M.

### A REMARK ON MULTIPLICATION MODULES

• Choi, Chang-Woo;Kim, Eun-Sup
• Bulletin of the Korean Mathematical Society
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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.

### ZERO-DIVISOR GRAPHS OF MULTIPLICATION MODULES

• Lee, Sang Cheol;Varmazyar, Rezvan
• Honam Mathematical Journal
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• v.34 no.4
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• pp.571-584
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• 2012
• In this study, we investigate the concept of zero-divisor graphs of multiplication modules over commutative rings as a natural generalization of zero-divisor graphs of commutative rings. In particular, we study the zero-divisor graphs of the module $\mathbb{Z}_n$ over the ring $\mathbb{Z}$ of integers, where $n$ is a positive integer greater than 1.

### ON ENDOMORPHISMS OF MULTIPLICATION MODULES

• CHOI, CHANG WOO;SMITH, PATRICK F.
• Journal of the Korean Mathematical Society
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• v.31 no.1
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• pp.89-95
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• 1994

### ON MULTIPLICATION MODULES

• Kim, Eun Sup;Choi, Chang Woo
• Kyungpook Mathematical Journal
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• v.32 no.1
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• pp.97-102
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• 1992