By Rotman J.J.

ISBN-10: 0821847414

ISBN-13: 9780821847411

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11 that ea Aea , a 2 Q0 , are local K-algebras. 13, we conclude that the algebras ea Aea , a 2 Q0 , are indecomposable. 3. The Jacobson radical 39 Let e be a central idempotent of KQ=I . 14. Therefore, e is of the form e D a20 ea , for a subset 0 of Q0 . ˛/ of ˛ belong simultaneously to 0 or Q0 n 0 . Hence, e is a nontrivial central idempotent of KQ=I if and only if Q is a disjoint union of two subquivers with the nonempty sets 0 and Q0 n 0 of vertices. 14, we conclude that the K-algebra KQ=I is indecomposable if and only if the quiver Q is connected.

Q; I /. Consider the K-linear homomorphism M M M fa W Ma ! M / ! N / is a homomorphism of right A-modules. Indeed, take x D xa 2 Ma and wN D w C I for a path w in Q with source a 2 Q0 and target b 2 Q0 . M / and wN 2 KQ=I , and our claim follows. M / belongs to mod A, because the quiver Q is finite. Q; I / ! Q; I / ! mod A. A standard checking shows that we have equivalences of functors GF ! 1Mod A and F G ! Q;I / . We note that the K-linear equivalence of categories F W Mod A ! Q; I /. 11. Let Q be a finite acyclic quiver, K a field and A D KQ.

11) that rad A is also the intersection of all maximal two-sided ideals of A. Let I be a two-sided ideal of a finite dimensional K-algebra A. For an integer m 1, we denote by I m the two-sided ideal of A consisting of all finite sums of elements of the form x1 x2 : : : xm , where x1 ; x2 ; : : : ; xm 2 I . The ideal I is said to be nilpotent if I m D 0 for some m 1. Observe that then every element x of I is nilpotent, that is, x r D 0 for some r 1. We will show that the Jacobson radical of a finite dimensional K-algebra is nilpotent.

### Advanced algebra by Rotman J.J.

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