• 호남수학학술지
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• 제43권3호
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• pp.403-416
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• 2021

It is known that the rank 2 stably free syzygy module P⁽²⁾ is not free. This algebraic fact was proved analytically, but this remarkable fact still lacks of a simple algebraic proof. The main purpose of this paper is to give a partially algebraic proof by making use of a theorem whose proof is quite topological, and the further properties of the module will be discussed.

• 대한수학회지
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• 제50권6호
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• pp.1271-1290
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• 2013

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-$m$-system sets, M-$n$-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "primeM-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and $Hom_R(M,X){\neq}0$ for all modules X in the category ${\sigma}[M]$, then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in ${\sigma}[M]$ and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.

• 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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• 제57권3호
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• pp.763-780
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• 2020

Let R be a commutative ring. An R-module M is said to be w-flat if Tor ^R₁ (M, N) is GV -torsion for any R-module N. It is known that every flat module is w-flat, but the converse is not true in general. The w-flat dimension of a module is defined in terms of w-flat resolutions. In this paper, we study the w-flat dimension of an injective w-module. To do so, we introduce and study the so-called w-copure (resp., strongly w-copure) flat modules and the w-copure flat dimensions for modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. We also study change of rings theorems for the w-copure flat dimension in various contexts. Finally some illustrative examples regarding the introduced concepts are given.

• 대한수학회지
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• 제54권4호
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• pp.1281-1299
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• 2017

A submodule N of a module M is called ${\mathcal{S}}$-closed (in M) if M/N is nonsingular. It is well-known that the class Closed of short exact sequences determined by closed submodules is a proper class in the sense of Buchsbaum. However, the class $\mathcal{S}-Closed$ of short exact sequences determined by $\mathcal{S}$-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class ${\langle}\mathcal{S-Closed}{\rangle}$ containing $\mathcal{S-Closed}$ in terms of $\mathcal{S}$-closed submodules. We show that this class coincides with the proper classes projectively generated by Goldie torsion modules and coprojectively generated by nonsingular modules. Moreover, for a right nonsingular ring R, it coincides with the proper class generated by neat submodules if and only if R is a right SI-ring. In abelian groups, the elements of this class are exactly torsionsplitting. In the second part, coprojective modules of this class which we call ec-flat modules are also investigated. We prove that injective modules are ec-flat if and only if each injective hull of a Goldie torsion module is projective if and only if every Goldie torsion module embeds in a projective module. For a left Noetherian right nonsingular ring R of which the identity element is a sum of orthogonal primitive idempotents, we prove that the class ${\langle}\mathcal{S-Closed}{\rangle}$ coincides with the class of pure-exact sequences of modules if and only if R is a two-sided hereditary, two-sided $\mathcal{CS}$-ring and every singular right module is a direct sum of finitely presented modules.

• 대한수학회지
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• 제57권6호
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• pp.1389-1406
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• 2020

Let Λ be an Artin algebra and mod Λ the category of finitely generated right Λ-modules. We prove that the radical layer length of Λ is an upper bound for the radical layer length of mod Λ. We give an upper bound for the extension dimension of mod Λ in terms of the injective dimension of a certain class of simple right Λ-modules and the radical layer length of DΛ.

• 충청수학회지
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• 제15권2호
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• pp.35-39
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• 2003

In this paper we study a ring over which every left module of finite length has an injective hull of finite length. We consider a ring that is a finite intermediate normalizing extension ring of such a ring. We also consider the subrings of such a ring.

• 대한수학회지
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• 제38권5호
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• pp.987-1030
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• 2001

Some discrimination of modules whose endomorhism rings are prime is introduced, by means of structures of submodules inducing prime ideals of the endomorphism ring End(sub)R (M) of a left R-module (sub)RM over a ring R. Modules with non-prime endomorphism rings are contrapositively studied as well.

• 대한수학회지
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• 제52권4호
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• pp.821-838
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• 2015

We investigate the relative and Tate cohomology theories with respect to Ding modules and complexes, consider their relations with classical and Gorenstein cohomology theories. As an application, the Avramov-Martsinkovsky type exact sequence of Ding modules is obtained.

• 호남수학학술지
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• 제24권1호
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• pp.1-8
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• 2002

In this paper we will show that if E is an injective module over a commutative ring A, then the sequence of sets $Ass_{A}(A/I^{n})^{*(E)}),\;n{\in}N,$ is increasing and ultimately constant. Also we will obtain some results concerning the integral closure of ideals related to some modules.