Commun. Korean Math. Soc. 2006 Vol. 21, No. 1, 37-43 Printed March 1, 2006

Sangwon Park, Dera Shin Dong-A University, Dong-A University

Abstract : We prove that $M_1 \ \Ar{f} \ M_2$ is an injective representation of a quiver $Q= \bullet \to \bullet$ if and only if $M_1$ and $M_2$ are injective left $R$-modules, $M_1 \ \Ar{f} \ M_2$ is isomorphic to a direct sum of representation of the types $E_1 \ \to \ 0$ and $E_2 \ \Ar{id} \ E_2$ where $E_1$ and $E_2$ are injective left $R$-modules. Then, we generalize the result so that a representation $M_1 \Ar{f_1} M_2 \Ar{f_2} \cdots \Ar{f_{n-1}} M_n$ of a quiver $Q=\bullet \to \bullet \to \cdots \to \bullet$ 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.

Keywords : module, quiver, representation of quiver, injective representation of quiver