• Korean Journal of Mathematics
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• 제18권3호
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• pp.243-252
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• 2010

In this paper we extend the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left R-modules to the projective properties of representations of quiver $Q={\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\rightarrow}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. And we show $0{\rightarrow}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module if and only if $0{\rightarrow}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. Then we show if P is a projective left R-module then $R[x]\longrightarrow^{id}P[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules. We also show that if $L\longrightarrow^{id}L$ is a projective representation of $Q={\bullet}{\rightarrow}{\bullet}$ as R-module, then $L[x]\longrightarrow^{id}L[x]$ is a projective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ as $R[x]$-modules.

• 호남수학학술지
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• 제41권3호
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• pp.531-538
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• 2019

We introduce FI-${\oplus}$-supplemented modules as a proper generalization of ${\oplus}$-supplemented modules. We prove that; (1) every finite direct sum of FI-${\oplus}$-supplemented R-modules is an FI-${\oplus}$-supplemented R-module for any ring R ; (2) if every left R-module is FI-${\oplus}$-supplemented over a semilocal ring R, then R is left perfect; (3) if M is a finitely generated torsion-free uniform R-module over a commutative integrally closed domain such that every direct summand of M is FI-${\oplus}$-supplemented, then M is a direct sum of cyclic modules.

• Korean Journal of Mathematics
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• 제20권3호
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• pp.343-352
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• 2012

In this paper we show that the projective properties of representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ as left $R[x]$-modules. We show that if P is a projective left R-module then $0{\longrightarrow}0{\longrightarrow}P[x]$ is a projective representation of a quiver Q as $R[x]$-modules, but $P[x]{\longrightarrow}0{\longrightarrow}0$ is not a projective representation of a quiver Q as $R[x]$-modules, if $P{\neq}0$. And we show a representation $0{\longrightarrow}P[x]\longrightarrow^{id}P[x]$ of a quiver Q is a projective representation, if P is a projective left R-module, but $P[x]\longrightarrow^{id}P[x]{\longrightarrow}0$ is not a projective representation of a quiver Q as $R[x]$-modules, if $P{\neq}0$. Then we show a representation $P[x]\longrightarrow^{id}P[x]\longrightarrow^{id}P[x]$ of a quiver Q is a projective representation, if P is a projective left R-module.

• 대한수학회논문집
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• 제25권4호
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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.

• 대한수학회보
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• 제44권2호
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• pp.225-231
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• 2007

Given an injective envelope E of a left R-module M, there is an associative Galois group Gal$({\phi})$. Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope $E[x^{-1}]$ of an inverse polynomial module $M[x^{-1}]$ as a left R[x]-module and we can define an associative Galois group Gal$({\phi}[x^{-1}])$. In this paper we describe the relations between Gal$({\phi})$ and Gal$({\phi}[x^{-1}])$. Then we extend the Galois group of inverse polynomial module and can get Gal$({\phi}[x^{-s}])$, where S is a submonoid of $\mathbb{N}$ (the set of all natural numbers).

• East Asian mathematical journal
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• 제24권2호
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• pp.139-144
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• 2008

Given an injective envelope E of a left R-module M, there is an associative Galois group Gal($\phi$). Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope E[$x^{-1}$] of an inverse polynomial module M[$x^{-1}$] as a left R[x]-module and we can define an associative Galois group Gal(${\phi}[x^{-1}]$). In this paper we extend the Galois group of inverse polynomial module and can get Gal(${\phi}[x^{-s}]$), where S is a submonoid of $\mathds{N}$ (the set of all natural numbers).

• 대한수학회지
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• 제51권6호
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• pp.1305-1319
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• 2014

A submodule N of a right R-module M is called coneat if for every simple right R-module S, any homomorphism $N{\rightarrow}S$ can be extended to a homomorphism $M{\rightarrow}S$. M is called coneat-flat if the kernel of any epimorphism $Y{\rightarrow}M{\rightarrow}0$ is coneat in Y. It is proven that (1) coneat submodules of any right R-module are coclosed if and only if R is right K-ring; (2) every right R-module is coneat-flat if and only if R is right V -ring; (3) coneat submodules of right injective modules are exactly the modules which have no maximal submodules if and only if R is right small ring. If R is commutative, then a module M is coneat-flat if and only if $M^+$ is m-injective. Every maximal left ideal of R is finitely generated if and only if every absolutely pure left R-module is m-injective. A commutative ring R is perfect if and only if every coneat-flat module is projective. We also study the rings over which coneat-flat and flat modules coincide.

• Korean Journal of Mathematics
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• 제20권3호
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• pp.307-314
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• 2012

Let M be a left R-module and R be a ring with unity, and $S=\{0,2,3,4,{\ldots}\}$ be a submonoid. Then $M[x^{-s}]=\{a_0+a_2x^{-2}+a_3x^{-3}+{\cdots}+a_nx^{-n}{\mid}a_i{\in}M\}$ is an $R[x^s]$-module. In this paper we show some properties of $M[x^{-s}]$ as an $R[x^s]$-module. Let $f:M{\rightarrow}N$ be an R-linear map and $\overline{M}[x^{-s}]=\{a_2x^{-2}+a_3x^{-3}+{\cdots}+a_nx^{-n}{\mid}a_i{\in}M\}$ and define $N+\overline{M}[x^{-s}]=\{b_0+a_2x^{-2}+a_3x^{-3}+{\cdots}+a_nx^{-n}{\mid}b_0{\in}N,\;a_i{\in}M}$. Then $N+\overline{M}[x^{-s}]$ is an $R[x^s]$-module. We show that given a short exact sequence $0{\rightarrow}L{\rightarrow}M{\rightarrow}N{\rightarrow}0$ of R-modules, $0{\rightarrow}L{\rightarrow}M[x^{-s}]{\rightarrow}N+\overline{M}[x^{-s}]{\rightarrow}0$ is a short exact sequence of $R[x^s]$-module. Then we show $E_1+\overline{E_0}[x^{-s}]$ is not an injective left $R[x^s]$-module, in general.

• 대한수학회보
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• 제51권2호
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• pp.579-594
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• 2014

Let R be a ring and $Nil_*$(R) be the prime radical of R. In this paper, we say that a ring R is left $Nil_*$-coherent if $Nil_*$(R) is coherent as a left R-module. The concept is introduced as the generalization of left J-coherent rings and semiprime rings. Some properties of $Nil_*$-coherent rings are also studied in terms of N-injective modules and N-flat modules.

• 대한수학회지
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• 제46권1호
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• pp.99-111
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• 2009

A ring is called left P-coherent if every principal left ideal is finitely presented. Let M be a right R-module with the endomorphism ring S. We mainly study the P-coherence of S. It is shown that S is a left P-coherent ring if and only if the left annihilator $ann_S$(X) is a finitely generated left ideal of S for any M-cyclic submodule X of M if and only if every cyclically M-presented right R-module has an M-torsionfree preenvelope. As applications, we investigate when the endomorphism ring S is left PP or von Neumann regular.