• Honam Mathematical Journal
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• v.35 no.2
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• pp.173-178
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• 2013

In this paper, we give some properties on multiplication Dedekind modules.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1071-1083
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• 2023

The main motivation of this paper is to introduce and study the notions of strong Dedekind rings and semi-regular injective modules. Specifically, a ring R is called strong Dedekind if every semi-regular ideal is Q₀-invertible, and an R-module E is called a semi-regular injective module provided Ext¹_R(T, E) = 0 for every 𝓠-torsion module T. In this paper, we first characterize rings over which all semi-regular injective modules are injective, and then study the semi-regular injective envelopes of R-modules. Moreover, we introduce and study the semi-regular global dimensions sr-gl.dim(R) of commutative rings R. Finally, we obtain that a ring R is a DQ-ring if and only if sr-gl.dim(R) = 0, and a ring R is a strong Dedekind ring if and only if sr-gl.dim(R) ≤ 1, if and only if any semi-regular ideal is projective. Besides, we show that the semi-regular dimensions of strong Dedekind rings are at most one.

• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.463-472
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• 2018

A Q-module is a module in which every nonzero submodule of M is a finite product of primary submodules of M. This paper is devoted to study some properties of Dedekind modules and Q-modules.

• 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.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.427-440
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• 2017

In this paper, in terms of the notions of strongly copure projective modules and the copure projective dimension cpD(R) of a ring R were defined in [12], we show that a domain R has $cpD(R){\leq}1$ if and only if R is a Gorenstein Dedekind domain.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.521-537
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• 2023

A module M is called a simple-direct-injective module if, whenever A and B are simple submodules of M with A ≅ B and B is a direct summand of M, then A is a direct summand of M. Some new characterizations of these modules are proved. The structure of simple-direct-injective modules over a commutative Dedekind domain is fully determined. Also, some relevant counterexamples are indicated to show that a left simple-direct-injective ring need not be right simple-direct-injective.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.59-69
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• 2024

Let R be a commutative ring. An R-module E is said to be regular injective provided that Ext¹_R(R/I, E) = 0 for any regular ideal I of R. We first show that the class of regular injective modules have the hereditary property, and then introduce and study the regular injective dimension of modules and regular global dimension of rings. Finally, we give some homological characterizations of total rings of quotients and Dedekind rings.

• Kyungpook Mathematical Journal
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• v.26 no.2
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• pp.89-112
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• 1986

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.37-42
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• 2011

Let M be an R-module and S a semigroup. Our goal is to discuss zero-divisors of the semigroup module M[S]. Particularly we show that if M is an R-module and S a commutative, cancellative and torsion-free monoid, then the R[S]-module M[S] has few zero-divisors of size n if and only if the R-module M has few zero-divisors of size n and Property (A).