• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.21-30
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• 2007

We say a module $M_R$ a semicommutative module if for any $m{\in}M$ and any $a{\in}R$, $ma=0$ implies $mRa=0$. This paper gives various properties of reduced, Armendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and $M_R$ be a p.p.-module, then $M_R$ is a semicommutative module iff $M_R$ is an Armendariz module. For any ring R, R is semicommutative iff A(R, ${\alpha}$) is semicommutative. Let R be a reduced ring, it is shown that for number $n{\geq}4$ and $k=[n=2]$, $T^k_n(R)$ is semicommutative ring but $T^{k-1}_n(R)$ is not.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.643-657
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• 2022

Let R be a ring and S a multiplicative subset of R. An R-module T is called u-S-torsion (u-always abbreviates uniformly) provided that sT = 0 for some s ∈ S. The notion of u-S-exact sequences is also introduced from the viewpoint of uniformity. An R-module F is called u-S-flat provided that the induced sequence 0 → A ⊗_R F → B ⊗_R F → C ⊗_R F → 0 is u-S-exact for any u-S-exact sequence 0 → A → B → C → 0. A ring R is called u-S-von Neumann regular provided there exists an element s ∈ S satisfying that for any a ∈ R there exists r ∈ R such that sα = rα². We obtain that a ring R is a u-S-von Neumann regular ring if and only if any R-module is u-S-flat. Several properties of u-S-flat modules and u-S-von Neumann regular rings are obtained.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1221-1233
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• 2021

In this paper, we introduce and investigate a new class of modules that is closely related to the class of Noetherian modules. Let R be a commutative ring and M be an R-module. We say that M is an r-Noetherian module if every r-submodule of M is finitely generated. Also, we call the ring R to be an r-Noetherian ring if R is an r-Noetherian R-module, or equivalently, every r-ideal of R is finitely generated. We show that many properties of Noetherian modules are also true for r-Noetherian modules. Moreover, we extend the concept of weakly Noetherian rings to the category of modules and we characterize Noetherian modules in terms of r-Noetherian and weakly Noetherian modules. Finally, we use the idealization construction to give non-trivial examples of r-Noetherian rings that are not Noetherian.

• East Asian mathematical journal
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• v.17 no.2
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• pp.367-374
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• 2001

Northcott and McKerrow proved that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-1}]$ is an injective left R[x]-module. Park generalized Northcott and McKerrow's result so that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-S}]$ is an injective left $R[x^S]$-module, where S is a submonoid of $\mathbb{N}$($\mathbb{N}$ is the set of all natural numbers). In this paper we extend the injective property to the Laurent power series module so that if R is a ring and E is an injective left R-module, then $E[[x^{-1},x]]$ is an injective left $R[x^S]$-module.

• Communications of the Korean Mathematical Society
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• v.32 no.2
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• pp.277-285
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• 2017

In the various module generalizations of the concepts of prime (semiprime) for a ring, the question "when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated.

• Journal of the Korean Mathematical Society
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• v.44 no.3
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• pp.719-731
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• 2007

R is called a right ${\Pi}-coherent$ ring in case every finitely generated torsion less right R-module is finitely presented. In this paper, we define a dimension for rings, called ${\Pi}-coherent$ dimension, which measures how far away a ring is from being ${\Pi}-coherent$. This dimension has nice properties when the ring in question is coherent. In addition, we study some properties of ${\Pi}-coherent$ rings in terms of preenvelopes and precovers.

• Journal of applied mathematics & informatics
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• v.17 no.1_2_3
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• pp.771-778
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• 2005

In this paper we shall give some group properties derived from the characteristic ring-module $_X(M)$, using the fact that $_X(M)_H$ is a conjugate to $_X(M)_{Ha}$ when M is an invertible right R-module. Also we shall prove that_X(M)$ is group-isomorphic to TR and some normal subgroup properties if M is invertible and R is commutative.

• East Asian mathematical journal
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• v.16 no.1
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• pp.33-48
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• 2000

In this paper, for any ring R with an identity, in order to study prime ideals of the endomorphism ring $End_R$(M) of left R-module $_RM$, meet-prime submodules, prime radical, sum-prime submodules and the prime socle of a module are defined. Some relations of the prime radical, the prime socle of a module and the prime radical of the endomorphism ring of a module are investigated. It is revealed that meet-prime(or sum-prime) modules and semi-meet-prime(or semi-sum-prime) modules have their prime, semi-prime endomorphism rings, respectively.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.459-467
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• 2013

In this paper, Matlis injective modules are introduced and studied. It is shown that every R-module has a (special) Matlis injective preenvelope over any ring R and every right R-module has a Matlis injective envelope when R is a right Noetherian ring. Moreover, it is shown that every right R-module has an ${\mathcal{F}}^{{\perp}1}$-envelope when R is a right Noetherian ring and $\mathcal{F}$ is a class of injective right R-modules.

• Journal of the Korean Mathematical Society
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• v.51 no.6
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• pp.1305-1319
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• 2014

A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism $N{\rightarrow}S$ can be extended to a homomorphism $M{\rightarrow}S$. M is called coneat-flat if the kernel of any epimorphism $Y{\rightarrow}M{\rightarrow}0$ is coneat in Y. It is proven that (1) coneat submodules of any right R-module are coclosed if and only if R is right K-ring; (2) every right R-module is coneat-flat if and only if R is right V -ring; (3) coneat submodules of right injective modules are exactly the modules which have no maximal submodules if and only if R is right small ring. If R is commutative, then a module M is coneat-flat if and only if $M^+$ is m-injective. Every maximal left ideal of R is finitely generated if and only if every absolutely pure left R-module is m-injective. A commutative ring R is perfect if and only if every coneat-flat module is projective. We also study the rings over which coneat-flat and flat modules coincide.