International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 2, Pages 97-101
doi:10.1155/S0161171202108192
Projective representations of quivers
Department of Mathematics, Dong-A University, Pusan 604-714, Korea
Received 30 August 2001
Copyright © 2002 Sangwon Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.