Exercise 10, § 1

Assume that $k$ is an algebraically closed field. Let Q and $Q'$ be finite quivers whose every loop has length zero. Prove that the algebra $A_{Q'}$ is equivalent in the sense of Morita (A, VIII, §6) to the algebra $A_Q$ if and only if the quivers Q and $Q'$ are isomorphic. (Take for modules $M,N$ the simple $k$-modules $k(s)$, for $s\in$ Som(Q).)