• Kyungpook Mathematical Journal
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• v.53 no.3
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• pp.407-417
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• 2013

In this article, we characterize t-Pr$\ddot{u}$fer modules in the class of faithful multiplication modules. As a corollary, we also characterize Krull modules. Several properties of a $t$-invertible submodule of a faithful multiplication module are given.

• 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.

• Journal of the Chungcheong Mathematical Society
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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.

• Journal of the Korean Mathematical Society
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• v.42 no.5
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• pp.933-948
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• 2005

Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = 1M. Let M be a non-zero multiplication R-module. Then we prove the following: (1) there exists a bijection: N(M)$\bigcap$V(ann$\_{R}$(M))$\rightarrow$Spec$\_{R}$(M) and in particular, there exists a bijection: N(M)$\bigcap$Max(R)$\rightarrow$Max$\_{R}$(M), (2) N(M) $\bigcap$ V(ann$\_{R}$(M)) = Supp(M) $\bigcap$ V(ann$\_{R}$(M)), and (3) for every ideal I of R, The ideal $\theta$(M) = $\sum$$\_{m(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P $\in$ Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P$^{n}$M$\neq$P$^{n+1}$ for every positive integer n, then $\bigcap$$^{$\_{n=1}$(P$^{n}$ + ann$\_{R}$(M)) $\in$ V(ann$\_{R}$(M)) = Supp(M) $\subseteq$ N(M).

• Journal of the Korean Mathematical Society
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• v.60 no.2
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• pp.327-339
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• 2023

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. In this article we introduce a new class of ring, called S-multiplication rings which are S-versions of multiplication rings. An R-module M is said to be S-multiplication if for each submodule N of M, sN ⊆ JM ⊆ N for some s ∈ S and ideal J of R (see for instance [4, Definition 1]). An ideal I of R is called S-multiplication if I is an S-multiplication R-module. A commutative ring R is called an S-multiplication ring if each ideal of R is S-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and S-P IR. Moreover, we generalize some properties of multiplication rings to S-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.1
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• pp.53-59
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• 2017

In this article, we generalize the concepts of several classes of domains (which are related to a Dedekind domain) to a torsion-free module and it is shown that for a faithful multiplication module over an integral domain, we characterize Dedekind modules, cyclic submodule modules, and discrete valuation modules in terms of factorable modules and a sort of Euclidean algorithm.

• East Asian mathematical journal
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• v.27 no.1
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• pp.75-82
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• 2011

In this paper, we will introduce the concept of fuzzy mulitplication module. We will define a new operation called a product on th family of all fuzzy submodules of a fuzzy mulitplication module. We will define a fuzzy subset of the idealization ring R+M and find some relations with the product of fuzzy submodules and product of fuzzy ideals of the idealization ring R+M. Some properties of weakly fuzzy prime submoduels and fuzzy prime submodules which are de ned by T.K.Mukherjee M.K.Sen and D.Roy will be introduced. We will investigate some properties of fuzzy prime submodules of a fuzzy multiplication module.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1285-1307
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• 2019

For a multiplication R-module M we consider the Zariski topology in the set Spec (M) of prime submodules of M. We investigate the relationship between the algebraic properties of the submodules of M and the topological properties of some subspaces of Spec (M). We also consider some topological aspects of certain frames. We prove that if R is a commutative ring and M is a multiplication R-module, then the lattice Semp (M/N) of semiprime submodules of M/N is a spatial frame for every submodule N of M. When M is a quasi projective module, we obtain that the interval ${\uparrow}(N)^{Semp}(M)=\{P{\in}Semp(M){\mid}N{\subseteq}P\}$ and the lattice Semp (M/N) are isomorphic as frames. Finally, we obtain results about quantales and the classical Krull dimension of M.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.213-226
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• 2022

Let R be a ring and M an R-module. Then M is said to be regular w-flat provided that the natural homomorphism I ⊗_R M → R ⊗_R M is a w-monomorphism for any regular ideal I. We distinguish regular w-flat modules from regular flat modules and w-flat modules by idealization constructions. Then we give some characterizations of total quotient rings and Prüfer v-multiplication rings (PvMRs for short) utilizing the homological properties of regular w-flat modules.