Projective representations of quivers (Q1607922)

Summary: We prove that \(P_{1} @> f>> P_{2}\) is a projective representation of a quiver \(Q= \bullet \to \bullet\) if and only if \(P_{1}\) and \(P_{2}\) are projective left \(R\)-modules, \(f\) is an injection, and \(f (P_1)\subset P_2\) is a summand. Then, we generalize the result so that a representation \[ M_{1} @> {f_1}>> M_{2} @> {f_2}>> \cdots @> {f_{n-2}}>> M_{n-1} @> {f_{n-1}}>> M_{n} \] of a quiver \[ Q=\bullet \to \bullet \to \bullet \cdots \bullet \to \bullet \to \bullet \] is a projective representation if and only if each \(M_{i}\) is a projective left \(R\)-module and the representation is a direct sum of projective representations.