• 호남수학학술지
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• 제41권4호
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• pp.799-812
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• 2019

We introduce flat e-covers of modules and define e-perfect rings as a generalization of perfect rings. We prove that a ring is right perfect if and only if it is semilocal and right e-perfect which generalizes a result due to N. Ding and J. Chen. Moreover, in the light of the fact that a ring R is right perfect if and only if flat covers of any R-module are projective covers, we study on the rings over which flat covers of modules are (generalized) locally projective covers, and obtain some characterizations of (semi) perfect, A-perfect and B-perfect rings.

• 충청수학회지
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• 제20권4호
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• pp.569-575
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• 2007

Let M be a nonzero module. M has the property that every submodule of M lies over a direct summand of M. We study some properties of such a module. The endomorphism ring of such a module is also studied. The relationships of such a module to the semi-regular modules, and to the semi-perfect modules are described.

• 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.

• 대한수학회보
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• 제37권1호
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• pp.103-108
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• 2000

Using spectral sequences we calculate the highest nonvanishing index of Tor for modules of finite projective dimension. The result is applied to compute the depth of the highest nonvanishing Tor. This is one of the cases when a problem of Auslander is positive.

• Kyungpook Mathematical Journal
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• 제47권3호
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• pp.357-361
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• 2007

A right R-module P is $c$-separative provided that $$P{\oplus}P{{c}\atop{\simeq_-}}P{\oplus}Q{\Longrightarrow}P{\simeq_-}Q$$ for any right R-module Q. We get, in this paper, two sufficient conditions under which a right module is $c$-separative. A ring R is a hereditary ring provided that every ideal of R is projective. As an application, we prove that every projective right R-module over a hereditary ring is $c$-separative.

• 대한수학회보
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• 제60권6호
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• pp.1523-1537
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• 2023

Let R be a commutative ring with identity and S a multiplicative subset of R. First, we introduce and study the u-S-projective dimension and u-S-injective dimension of an R-module, and then explore the u-S-global dimension u-S-gl.dim(R) of a commutative ring R, i.e., the supremum of u-S-projective dimensions of all R-modules. Finally, we investigate u-S-global dimensions of factor rings and polynomial rings.

• 대한수학회보
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• 제60권6호
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• pp.1453-1461
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• 2023

Let R be a one-dimensional Noetherian domain with quotient field K and T be the integral closure of R in K. In this note we prove that if the conductor ideal (R :_K T) is a nonzero prime ideal, then every finitely generated reflexive (and hence finitely generated G-projective) R-module is isomorphic to a direct sum of some ideals.

• 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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• 제57권2호
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• pp.419-427
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• 2020

In this paper, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either finitely many minimal primes or finitely many maximal ideals then every finitely generated flat module over it is projective. It is also shown that if a particular subset of the prime spectrum of a ring satisfies some certain ascending or descending chain conditions, then every finitely generated flat module over this ring is projective. These results generalize some major results in the literature on the projectivity of finitely generated flat modules.

• Kyungpook Mathematical Journal
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• 제53권1호
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• pp.37-47
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• 2013

Let M and X be right R-modules. We introduce several modules relative to the class of B(M, X) and we investigate relation among these modules. In this note, we show if M is X-${\oplus}$-supplemented such that $M=M_1{\oplus}M_2$ implies $M_1$ and $M_2$ are relatively B-projective, then M is an X-H-supplemented module.