• Communications of the Korean Mathematical Society
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• v.11 no.2
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• pp.285-294
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• 1996

In this paper, we show relations between injective covers and direct sums, some commutative properties, and composition properties in the injective covers.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.261-267
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• 1995

Using the dual of a categorical definition of an injective envelope, Enochs defined an injective cover. In this paper we will show how injective covers can be used to characterize several well known classes of rings.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.589-596
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• 2007

In this paper we prove that the extension of pure injective module is pure injective if and only if the cotorsion envelope and the pure injective envelope of any R-module M are isomorphic over M. And we prove that if the product of pure injective envelopes of flat modules is a pure injective envelope and the product of flat covers is a flat cover, then the product of cotorsion envelopes is a cotorsion envelope.

• Bulletin of the Korean Mathematical Society
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• v.47 no.3
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• pp.611-622
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• 2010

A ring R is called left max-coherent provided that every maximal left ideal is finitely presented. $\mathfrak{M}\mathfrak{I}$ (resp. $\mathfrak{M}\mathfrak{F}$) denotes the class of all max-injective left R-modules (resp. all max-flat right R-modules). We prove, in this article, that over a left max-coherent ring every right R-module has an $\mathfrak{M}\mathfrak{F}$-preenvelope, and every left R-module has an $\mathfrak{M}\mathfrak{I}$-cover. Furthermore, it is shown that a ring R is left max-injective if and only if any left R-module has an epic $\mathfrak{M}\mathfrak{I}$-cover if and only if any right R-module has a monic $\mathfrak{M}\mathfrak{F}$-preenvelope. We also give several equivalent characterizations of MI-injectivity and MI-flatness. Finally, $\mathfrak{M}\mathfrak{I}$-dimensions of modules and rings are studied in terms of max-injective modules with the left derived functors of Hom.

• Bulletin of the Korean Mathematical Society
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• v.40 no.1
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• pp.167-176
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• 2003

In [3］, Del Valle, Enochs and Martinez studied flat envelopes over rings and they showed that over rings as in the title these are very well behaved. If we replace flat with injective and envelope with the dual notion of a cover we then have the injective covers. In this article we show that these injective covers over the commutative noetherian rings with global dimension at most 2 have properties analogous to those of the flat envelopes over these rings.

• Communications of the Korean Mathematical Society
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• v.16 no.4
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• pp.567-572
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• 2001

Wurful gave a characterization of those rings R which satisfy that for every ring extension $R{\subset}S$. Ho $m_{R}$(S, -) preserves injective envelopes. In this note, we consider an analogous problem concerning injective covers.

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1051-1066
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• 2013

By utilizing known characterizations of ${\omega}$-Noetherian rings in terms of injective modules, we give more characterizations of ${\omega}$-Noetherian rings. More precisely, we show that a commutative ring R is ${\omega}$-Noetherian if and only if the direct limit of GV -torsion-free injective R-modules is injective; if and only if every R-module has a GV -torsion-free injective (pre)cover; if and only if the direct sum of injective envelopes of ${\omega}$-simple R-modules is injective; if and only if the essential extension of the direct sum of GV -torsion-free injective R-modules is the direct sum of GV -torsion-free injective R-modules; if and only if every $\mathfrak{F}_{w,f}(R)$-injective ${\omega}$-module is injective; if and only if every GV-torsion-free R-module admits an $i$-decomposition.

• Communications of the Korean Mathematical Society
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• v.25 no.4
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• pp.497-510
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• 2010

Let R be a ring and n a fixed non-negative integer. $\cal{TI}_n$ (resp. $\cal{TF}_n$) denotes the class of all right R-modules of FGT-injective dimensions at most n (resp. all left R-modules of FGT-flat dimensions at most n). We prove that, if R is a right $\prod$-coherent ring, then every right R-module has a $\cal{TI}_n$-cover and every left R-module has a $\cal{TF}_n$-preenvelope. A right R-module M is called n-TI-injective in case $Ext^1$(N,M) = 0 for any $N\;{\in}\;\cal{TI}_n$. A left R-module F is said to be n-TI-flat if $Tor_1$(N, F) = 0 for any $N\;{\in}\;\cal{TI}_n$. Some properties of n-TI-injective and n-TI-flat modules and their relations with $\cal{TI}_n$-(pre)covers and $\cal{TF}_n$-preenvelopes are also studied.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1457-1482
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• 2017

Let S and R be rings and $_SC_R$ a (faithfully) semidualizing bimodule. We introduce and study C-weak flat and C-weak injective modules as a generalization of C-flat and C-injective modules ([21]) respectively, and use them to provide additional information concerning the important Foxby equivalence between the subclasses of the Auslander class ${\mathcal{A}}_C$ (R) and that of the Bass class ${\mathcal{B}}_C$ (S). Then we study the stability of Auslander and Bass classes, which enables us to give some alternative characterizations of the modules in ${\mathcal{A}}_C$ (R) and ${\mathcal{B}}_C$ (S). Finally we consider an open question which is closely relative to the main results ([11]), and discuss the relationship between the Bass class ${\mathcal{B}}_C$(S) and the class of Gorenstein injective modules.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.11-24
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• 2013

Let m and n be fixed positive integers and M a right R-module. Recall that M is said to be ($m$, $n$)-injective if $Ext^1$(P, M) = 0 for any ($m$, $n$)-presented right R-module P; M is said to be ($m$, $n$)-flat if $Tor_1$(N, P) = 0 for any ($m$, $n$)-presented left R-module P. In terms of some derived functors, relative injective or relative flat resolutions and dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and p.p. rings are given.