Projective representations of quivers

In the first part of this article, we describe the projective representations in the category. of representations by, modules of a quiver which does not contain any, cycles and the quiver A(infinity) as a subquiver, that is, the so-called rooted quivers. As a consequence of this, we show when the category, of representations by, modules of a quiver admits projective covers. In the second part, we develop a technique involving matrix computations for the quiver. A(infinity), which will allow us to characterize the projective representations of A(infinity) This it will improve some previous results and make more accurate the statement made in Benson (1991). We think this technique can he applied in many other general situations to provide information about the decomposition of a projective module.