INJECTIVE REPRESENTATIONS OF QUIVERS

Title & Authors
INJECTIVE REPRESENTATIONS OF QUIVERS
Park, Sang-Won; Shin, De-Ra;

Abstract
We prove that $\small{M_1\longrightarrow^f\;M_2}$ is an injective representation of a quiver $\small{Q={\bullet}{\rightarrow}{\bullet}}$ if and only if $\small{M_1\;and\;M_2}$ are injective left R-modules, $\small{M_1\longrightarrow^f\;M_2}$ is isomorphic to a direct sum of representation of the types $\small{E_l{\rightarrow}0}$ and $\small{M_1\longrightarrow^{id}\;M_2}$ where $\small{E_l\;and\;E_2}$ are injective left R-modules. Then, we generalize the result so that a representation$\small{M_1\longrightarrow^{f_1}\;M_2\; \longrightarrow^{f_2}\;\cdots\;\longrightarrow^{f_{n-1}}\;M_n}$ of a quiver $\small{Q={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\cdots}{\rightarrow}{\bullet}}$ is an injective representation if and only if each $\small{M_i}$ is an injective left R-module and the representation is a direct sum of injective representations.
Keywords
module;quiver;representation of quiver;injective representation of quiver;
Language
English
Cited by
1.
PROJECTIVE PROPERTIES OF REPRESENTATIONS OF A QUIVER OF THE FORM $Q={\bullet}\,^{\rightarrow}_{\rightarrow}\,{\bullet}{\rightarrow}{\bullet}$,;;

Korean Journal of Mathematics, 2009. vol.17. 4, pp.429-436
2.
PROJECTIVE AND INJECTIVE PROPERTIES OF REPRESENTATIONS OF A QUIVER $Q ={\bullet}{\rightarrow}{\bullet}{\rightarrow}{\bullet}$,;;

Korean Journal of Mathematics, 2009. vol.17. 3, pp.271-281
3.
PROJECTIVE PROPERTIES OF REPRESENTATIONS OF A QUIVER Q = • → • AS R[x]-MODULES,;;;

Korean Journal of Mathematics, 2010. vol.18. 3, pp.243-252
1.
PROJECTIVE REPRESENTATIONS OF A QUIVER WITH THREE VERTICES AND TWO EDGES AS R[x]-MODULES, Korean Journal of Mathematics, 2012, 20, 3, 343
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