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Proof. Assume that h : P → M is a projective cover of M and let g : N → P be a homomorphism such that hg is surjective. It follows that Im g + Ker h = P and therefore g is surjective, because by assumption Ker h is superﬂuous in P . This shows the suﬃciency. Conversely, assume that h : P → M has the stated property. Let N be a submodule of P such that N + Ker h = P . If g : N → P is the natural inclusion, then hg : N → M is surjective. Hence, by hypothesis, g is surjective. This shows that Ker h is superﬂuous and ﬁnishes the proof.
An idempotent e ∈ A is primitive if and only if the algebra eAe ∼ = End eA has only two idempotents 0 and e, that is, the algebra eAe is local. 8. Corollary. Let A be an arbitrary K-algebra and M a right Amodule. (a) If the algebra End M is local, then M is indecomposable. 4. Direct sum decompositions 23 (b) If M is finite dimensional and indecomposable, then the algebra End M is local and any A-module endomorphism of M is nilpotent or is an isomorphism. Proof. (a) If M decomposes as M = X1 ⊕ X2 with both X1 and X2 nonzero, then there exist projections pi : M → Xi and injections ui : Xi → M (for i = 1, 2) such that u1 p1 + u2 p2 = 1M .
6)). The idea of such a graphical representation seems to go back to the late forties (see Gabriel , Grothendieck , and Thrall ) but it became widespread in the early seventies, mainly due to Gabriel , . In an explicit form, the notions of quiver and linear representation of quiver were introduced by Gabriel in . It was the starting point of the modern representation theory of associative algebras. 1. Quivers and path algebras This ﬁrst section is devoted to deﬁning the graphical structures we are interested in and introducing the related terminology.