• 대한수학회논문집
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• 제21권1호
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• pp.37-43
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• 2006

We prove that $M_1\longrightarrow^f\;M_2$ is an injective representation of a quiver $Q={\bullet}{\rightarrow}{\bullet}$ if and only if $M_1\;and\;M_2$ are injective left R-modules, $M_1\longrightarrow^f\;M_2$ is isomorphic to a direct sum of representation of the types $E_l{\rightarrow}0$ and $M_1\longrightarrow^{id}\;M_2$ where $E_l\;and\;E_2$ are injective left R-modules. Then, we generalize the result so that a representation$M_1\longrightarrow^{f_1}\;M_2\; \longrightarrow^{f_2}\;\cdots\;\longrightarrow^{f_{n-1}}\;M_n$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\cdots}{\rightarrow}{\bullet}$ is an injective representation if and only if each $M_i$ is an injective left R-module and the representation is a direct sum of injective representations.

• Korean Journal of Mathematics
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• 제17권3호
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• pp.271-281
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• 2009

We define injective and projective representations of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$. Then we show that a representation $M_1\longrightarrow[50]^{f1}M_2\longrightarrow[50]^{f2}M_3$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ is projective if and only if each $M_1,\;M_2,\;M_3$ is projective left R-module and $f_1(M_1)$ is a summand of $M_2$ and $f_2(M_2)$ is a summand of $M_3$. And we show that a representation $M_1\longrightarrow[50]^{f1}M_2\longrightarrow[50]^{f2}M_3$ of a quiver $Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$ is injective if and only if each $M_1,\;M_2,\;M_3$ is injective left R-module and $ker(f_1)$ is a summand of $M_1$ and $ker(f_2)$ is a summand of $M_2$.

• 대한수학회보
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• 제50권2호
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• pp.389-398
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• 2013

Let R be a ring and $\mathcal{Q}$ be a quiver. In this paper we give another definition of purity in the category of quiver representations. Under such definition we prove that the class of all pure injective representations of $\mathcal{Q}$ by R-modules is preenveloping. In case $\mathcal{Q}$ is a left rooted semi-co-barren quiver and R is left Noetherian, we show that every cotorsion flat representation of $\mathcal{Q}$ is pure injective. If, furthermore, R is $n$-perfect and $\mathcal{F}$ is a flat representation $\mathcal{Q}$, then the pure injective dimension of $\mathcal{F}$ is at most $n$.

• Korean Journal of Mathematics
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• 제16권3호
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• pp.323-334
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• 2008

We define injective and projective representations of quivers with two vertices with n arrows. In the representation of quivers we denote n edges between two vertices as ${\Rightarrow}$ and n maps as $f_1{\sim}f_n$, and $E{\oplus}E{\oplus}{\cdots}{\oplus}E$ (n times) as ${\oplus}_nE$. We show that if E is an injective left R-module, then $${\oplus}_nE{\Longrightarrow[50]^{p_1{\sim}p_n}}E$$ is an injective representation of $Q={\bullet}{\Rightarrow}{\bullet}$ where $p_i(a_1,a_2,{\cdots},a_n)=a_i,\;i{\in}\{1,2,{\cdots},n\}$. Dually we show that if $M_1{\Longrightarrow[50]^{f_1{\sim}f_n}}M_2$ is an injective representation of a quiver $Q={\bullet}{\Rightarrow}{\bullet}$ then $M_1$ and $M_2$ are injective left R-modules. We also show that if P is a projective left R-module, then $$P\Longrightarrow[50]^{i_1{\sim}i_n}{\oplus}_nP$$ is a projective representation of $Q={\bullet}{\Rightarrow}{\bullet}$ where $i_k$ is the kth injection. And if $M_1\Longrightarrow[50]^{f_1{\sim}f_n}M_2$ is an projective representation of a quiver $Q={\bullet}{\Rightarrow}{\bullet}$ then $M_1$ and $M_2$ are projective left R-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.

• 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.