  For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the
projective representations by computing the multiplicity of a given
indecomposable projective in the tensor product of two indecomposable
projectives. Motivated by this problem for arbitrary representations, we study
idempotents in the representation ring of Q (the free abelian group on the
indecomposable representations, with multiplication given by tensor product).
We give a general technique for constructing such idempotents and for
decomposing the representation ring into a direct product of ideals, utilizing
morphisms between quivers and categorical Moebius inversion.
