• Journal of the Korean Mathematical Society
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• v.46 no.1
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• pp.51-58
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• 2009

Let R be a ring. A right R-module M is called perfect if M possesses a projective cover. In this paper, we consider the relationship between the class of perfect modules and other classes of modules. Some known rings are characterized by these relationships.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.141-154
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• 2022

Let R be a commutative ring with identity. Denote by w𝒫_w the class of weak w-projective R-modules and by w𝒫_w^⊥ the right orthogonal complement of w𝒫_w. It is shown that (w𝒫_w, w𝒫_w^⊥) is a hereditary and complete cotorsion theory, and so every R-module has a special weak w-projective precover. We also give some necessary and sufficient conditions for weak w-projective modules to be w-projective. Finally it is shown that when we discuss the existence of a weak w-projective cover of a module, it is enough to consider the w-envelope of the module.

• Honam Mathematical Journal
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• v.41 no.4
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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.

• Honam Mathematical Journal
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• v.42 no.3
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• pp.591-600
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• 2020

A module M is called ss-semilocal if every submodule U of M has a weak supplement V in M such that U∩V is semisimple. In this paper, we provide the basic properties of ss-semilocal modules. In particular, it is proved that, for a ring R, _RR is ss-semilocal if and only if every left R-module is ss-semilocal if and only if R is semilocal and Rad(R) ⊆ Soc(RR). We define projective ss-covers and prove the rings with the property that every (simple) module has a projective ss-cover are ss-semilocal.

• Journal for History of Mathematics
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• v.11 no.2
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• pp.83-90
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• 1998

Observing that for any compact space X, the minimal basically disconnected cover ${\bigwedge}Λ_X$ : ${\bigwedge}Λ_X{\leftrightarro}$ is ${\sigma}Z^#$-irreducible, we will show that the projective objects in the category of compact spaces and ${\sigma}Z^#$-irreducible maps are precisely basically disconnected spaces.

• Communications of the Korean Mathematical Society
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• v.1 no.1
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• pp.99-101
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• 1986

• Kyungpook Mathematical Journal
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• v.31 no.2
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• pp.201-205
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• 1991

• The Pure and Applied Mathematics
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• v.9 no.1
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• pp.9-17
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• 2002

In this Paper, we will show that every basically disconnected space is a projective object in the category $Tych_{\sigma}$ of Tychonoff spaces and $_{\sigma}Z^{#}$ -irreducible maps and that if X is a space such that ${\Beta} {\Lambda} X={\Lambda} {\Beta} X$, then X has a projective cover in $Tych_{\sigma}$. Moreover, observing that for any weakly Linde1of space, ${\Lambda} X : {\Lambda} X\;{\longrightarrow}\;X$ is $_{\sigma}Z^{#}$-irreducible, we will show that the projective objects in $wLind_{\sigma}$/ of weakly Lindelof spaces and $_{\sigma}Z^{#}$-irreducible maps are precisely the basically disconnected spaces.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.143-154
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• 2008

Lomp [9] has studied finitely generated projective modules over semilocal rings. He obtained the following: finitely generated projective modules over semilocal rings are semilocal. We shall give necessary and sufficient conditions for finitely generated modules to be semilocal modules. By using a lifting property, we also give characterizations of right perfect (semiperfect) rings. Our main results can be summarized as follows: (1) Let M be a finitely generated module. Then M has finite hollow dimension if and only if M is weakly supplemented if and only if M is semilocal. (2) A ring R is right perfect if and only if every flat right R-module is lifting and every right R-module has a flat cover if and only if every quasi-projective right R-module is lifting. (3) A ring R is semiperfect if and only if every finitely generated flat right R-module is lifting if and only if RR satisfies the lifting property for simple factor modules.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1619-1625
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• 2017

The aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic.