• Honam Mathematical Journal
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• v.31 no.4
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• pp.529-540
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• 2009

The authors have investigated several properties of endomorphism rings of modules. In particular, a closedly simple module and a closedly semi-simple module were defined and then the regularities of its rings of endomorphisms were discussed.

• 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.

• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.651-663
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• 2008

In this paper, we first study some characterizations of left MC2 rings. Next, by introducing left nil-injective modules, we discuss and generalize some well known results for a ring whose simple singular left modules are Y J-injective. Finally, as a byproduct of these results we are able to show that if R is a left MC2 left Goldie ring whose every simple singular left R-module is nil-injective and GJcp-injective, then R is a finite product of simple left Goldie rings.

• Korean Journal of Mathematics
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• v.7 no.2
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• pp.215-218
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• 1999

We investigate characterizations of rings whose simple singular right R-modules are GP-injective. It is proved that if R is a semiprime ring whose simple singular right R-modules are GP-injective, then the center $Z(R)$ of R is a von Neumann regular ring. We consider the condition ($^*$): R satisfies $l(a){\subseteq}r(a)$ for any $a{\in}R$. Also it is shown that if R satisfies ($^*$) and every simple singular right R-module is GP-injective, then R is a reduced weakly regular ring.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.549-556
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• 2015

In this paper, we characterize w-Noetherian modules in terms of polynomial modules and w-Nagata modules. Then it is shown that for a finite type w-module M, every w-epimorphism of M onto itself is an isomorphism. We also define and study the concepts of w-Artinian modules and w-simple modules. By using these concepts, it is shown that for a w-Artinian module M, every w-monomorphism of M onto itself is an isomorphism and that for a w-simple module M, $End_RM$ is a division ring.

• Bulletin of the Korean Mathematical Society
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• v.42 no.1
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• pp.87-92
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• 2005

Let A be a polynomial ring over a field and M a simple A-module. We generalize one result of Song about the description of the injective envelope $E_A$ (M) in terms of modules of generalized fractions.

• Communications of the Korean Mathematical Society
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• v.18 no.4
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• pp.615-620
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• 2003

Semi-meet-prime projective modules and fully invariant meet-prime submodules of a projective module are studied. Actually a generalization of the Schur's lemma and semi-prime endmorphism rings of projective modules are considered.

• Journal of the Korean Data and Information Science Society
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• v.18 no.3
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• pp.803-808
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• 2007

In present article, we consider the embeddability problems for finite dimensional irreducible modules over a complex simple Lie algebra L. For s1(n,C) modules, we determine when one can be embedded into the other if s1(n,C) modules are tensor products of fundamental modules.

• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1687-1695
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• 2023

In this paper, we study the Green ring r(𝔴⁰_n) of the weak Hopf algebra 𝔴⁰_n based on Taft Hopf algebra H_n(q). Let R(𝔴⁰_n) := r(𝔴⁰_n) ⊗_ℤ ℂ be the Green algebra corresponding to the Green ring r(𝔴⁰_n). We first determine all finite dimensional simple modules of the Green algebra R(𝔴⁰_n), which is based on the observations of the roots of the generating relations associated with the Green ring r(𝔴⁰_n). Then we show that the nilpotent elements in r(𝔴⁰_n) can be written as a sum of finite dimensional indecomposable projective 𝔴⁰_n-modules. The Jacobson radical J(r(𝔴⁰_n)) of r(𝔴⁰_n) is a principal ideal, and its rank equals n - 1. Furthermore, we classify all finite dimensional non-simple indecomposable R(𝔴⁰_n)-modules. It turns out that R(𝔴⁰_n) has n² - n + 2 simple modules of dimension 1, and n non-simple indecomposable modules of dimension 2.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1549-1557
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• 2015

Characterizations of closed subgroups in abelian groups have been generalized to modules in essentially dierent ways; they are in general inequivalent. Here we consider the relations between these generalizations over commutative rings, and we characterize the commutative rings over which they coincide. These are exactly the commutative noetherian distributive rings. We also give a characterization of c-injective modules over commutative noetherian distributive rings. For a noetherian distributive ring R, we prove that, (1) direct product of simple R-modules is c-injective; (2) an R-module D is c-injective if and only if it is isomorphic to a direct summand of a direct product of simple R-modules and injective R-modules.