INJECTIVE REPRESENTATIONS OF QUIVERS

Abstract

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