• 대한수학회보
• /
• 제32권2호
• /
• pp.321-328
• /
• 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]).

• Korean Journal of Mathematics
• /
• 제20권3호
• /
• pp.315-321
• /
• 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.

• 대한수학회논문집
• /
• 제13권4호
• /
• pp.727-733
• /
• 1998

In this short paper we shall find some properties on multiplication modules and prove three theorems.

• 충청수학회지
• /
• 제25권4호
• /
• pp.717-723
• /
• 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.

• 대한수학회보
• /
• 제35권1호
• /
• pp.33-43
• /
• 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.

• 대한수학회보
• /
• 제56권1호
• /
• pp.83-102
• /
• 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.

• 호남수학학술지
• /
• 제27권1호
• /
• pp.1-8
• /
• 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.

• 대한수학회보
• /
• 제31권2호
• /
• pp.163-165
• /
• 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.

• 호남수학학술지
• /
• 제39권2호
• /
• pp.275-296
• /
• 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.

• 대한수학회보
• /
• 제42권1호
• /
• pp.121-131
• /
• 2005

This paper is motivated by the results in [6]. We study some properties of strongly irreducible submodules of a module. In fact, our objective is to investigate strongly irreducible modules and to examine in particular when sub modules of a module are strongly irreducible. For example, we show that prime submodules of a multiplication module are strongly irreducible, and a characterization is given of a multiplication module over a Noetherian ring which contain a non-prime strongly irreducible submodule.