• 대한수학회보
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• 제35권1호
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• pp.33-43
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• 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
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• 제51권3호
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• pp.613-619
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• 2014

It is well known that a direct sum of CLS-modules is not, in general, a CLS-module. It is proved that if $M=M_1{\oplus}M_2$, where $M_1$ and $M_2$ are CLS-modules such that $M_1$ and $M_2$ are relatively ojective (or $M_1$ is $M_2$-ejective), then M is a CLS-module and some known results are generalized.

• 대한수학회보
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• 제46권6호
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• pp.1069-1077
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• 2009

In 1984, Oshiro [11] has studied the decomposition of continuous lifting modules. He obtained the following: every continuous lifting module has an indecomposable decomposition. In this paper, we study extending lifting modules. We show that every extending lifting module has an indecomposable decomposition. This result is an expansion of Oshiro's result mentioned above. And we consider some application of this result.

• Kyungpook Mathematical Journal
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• 제45권3호
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• pp.445-453
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• 2005

We say that a module M is weak lifting if M is supplemented and every supplement submodule of M is a direct summand. The module M is called lifting, if it is weak lifting and amply supplemented. This paper investigates the structure of weak lifting modules and lifting modules having small radical over commutative noetherian rings.

• Kyungpook Mathematical Journal
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• 제59권3호
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• pp.391-401
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• 2019

A module is said to be ${\pi}$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this article, we focus on the class of modules having the ${\pi}$-extending property by looking at the singularity of quotient submodules. By doing so, we provide counterexamples, using hypersurfaces in projective spaces over complex numbers, to show that being generalized ${\pi}$-extending is not inherited by direct summands. Moreover, it is shown that the direct sums of generalized ${\pi}$-extending modules are generalized ${\pi}$-extending.

• 대한수학회보
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• 제58권1호
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• pp.147-161
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• 2021

In this paper, we introduced and studied sa-supplement submodules. A submodule U of a module V is called an sa-supplement submodule in V if there exists a submodule T of V such that V = T + U and U ∩ T is semiartinian. The class of sa-supplement sequences ������ is a proper class which is generated by socle-free modules injectively. We studied modules that have an sa-supplement in every extension, modules whose all submodules are sa-supplement and modules whose all sa-supplement submodules are direct summand. We provided new characterizations of right semiartinian rings and right SSI rings.

• Korean Journal of Mathematics
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• 제22권1호
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• pp.151-167
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• 2014

Let M, N be modules over a ring R and $[M,N]=Hom_R(M,N)$. The concern is study of: (1) Some fundamental properties of [M, N] when [M, N] is regular or semipotent. (2) The substructures of [M, N] such as radical, the singular and co-singular ideals, the total and others has raised new questions for research in this area. New results obtained include necessary and sufficient conditions for [M, N] to be regular or semipotent. New substructures of [M, N] are studied and its relationship with the Tot of [M, N]. In this paper we show that, the endomorphism ring of a module M is regular if and only if the module M is semi-injective (projective) and the kernel (image) of every endomorphism is a direct summand.

• 대한수학회지
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• 제53권2호
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• pp.403-414
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• 2016

Let R be a ring, and let M be a left R-module. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Any finite direct sum of Rad-supplementing modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal, reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is Rad-supplementing if and only if R is reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is left perfect, where P(R) is the sum of all left ideals I of R such that Rad I = I.

• 대한수학회보
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• 제45권1호
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• pp.59-66
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• 2008

Keskin and Harmanci defined the family B(M,X) = ${A{\leq}M|{\exists}Y{\leq}X,{\exists}f{\in}Hom_R(M,X/Y),\;Ker\;f/A{\ll}M/A}$. And Orhan and Keskin generalized projective modules via the class B(M, X). In this note we introduce X-local summands and X-hollow modules via the class B(M, X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module P contains Rad(P), then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang's result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with $K{\in}B$(H, X), if $H{\oplus}H$ has the internal exchange property, then H has a local endomorphism ring.

• 대한수학회보
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• 제52권5호
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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.