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Therefore, if i ∈ I and j ∈ J, we have ei Aej = ei Acej = ei cAej = 0 and similarly ej Aei = 0. 7. Lemma. Let Q be a finite quiver. The path algebra KQ is connected if and only if Q is a connected quiver. Proof. Assume that Q is not connected and let Q be a connected component of Q. Denote by Q the full subquiver of Q having as set of points Q0 = Q0 \Q0 . By hypothesis, neither Q nor Q is empty. Let a ∈ Q0 and b ∈ Q0 . Because Q is not connected, an arbitrary path w in Q is entirely contained in either Q or (a connected component of) Q .

Similarly, if r is a retraction of u, then r is surjective, u is injective, u is a section of r, and there exist direct sum decompositions M = Im u ⊕ Ker r ∼ = L ⊕ Ker r. An A-homomorphism h : M → N is called a section (or a retraction) if h admits a retraction (or a section, respectively). hn−1 hn+1 h n A sequence · · · −→ Xn−1 −→ Xn −→X n+1 −→Xn+2 −→ · · · (inﬁnite or ﬁnite) of right A-modules connected by A-homomorphisms is called exact if Ker hn = Im hn−1 for any n. In particular u r 0 −→ L−→M −→N −→ 0 is called a short exact sequence if u is a monomorphism, r is an epimorphism and Ker r = Im u.

Let A = M2 (K) and let e1 = (10 00 ), e2 = (00 01 ). Then e1 , e2 are primitive orthogonal idempotents of A such that 1A = e1 + e2 and AA = e1 A ⊕ e2 A. The algebra A is semisimple, S(1) = P (1) = I(1) ∼ = S(2) = P (2) = I(2) and dimK S(1) = dimK S(2) = 2. I. 6. Basic algebras and embeddings of module categories Throughout, we need essentially the following class of algebras (see [73], [125], and [131] for historical notes). 1. Definition. Assume that A is a K-algebra with a complete set {e1 , .