Projective representations of quivers

Abstract

We prove that P1 →f P2 is a projective representation of a quiver Q=•→• if and only if P1 and P2 are projective left R-modules, f is an injection, and f (P 1)⊂P 2 is a summand. Then, we generalize the result so that a representation M1 →f1  M2  →f2⋯→fn−2  Mn−1→fn−1  Mn of a quiver Q=•→•→•⋯•→•→• is projective representation if and only if each Mi is a projective left R-module and the representation is a direct sum of projective representations.