MA3203 Ring theory Spring 2019 Norwegian University of Science and Technology Exercise set 3 Department of Mathematical Sciences

1 Decompose the following representation into indecomposable representations: 1 1 1 1 1 1 1 1 2 2 2 k k k .

α γ 2 Let Q be the quiver 1 2 3 and Λ = kQ. β δ

a) Find the representations corresponding to the modules Λei, for i = 1, 2, 3. Let R be a ring and M an R-module. The endomorphism ring is defined as EndR(M) := {f : M → M | f R-homomorphism}.

b) Show that EndR(M) is indeed a ring. c) Show that an R-module M is decomposable if and only if its endomorphism ring EndR(M) contains a non-trivial idempotent, that is, an element f 6= 0, 1 such that f 2 = f. ∼ Hint: If M = M1 ⊕ M2 is decomposable, consider the projection morphism M → M1. Conversely, any idempotent can be thought of as a projection mor- phism.

d) Using c), show that the representation corresponding to Λe1 is indecomposable. ∼ Hint: Prove that EndΛ(Λe1) = k by showing that every Λ-homomorphism Λe1 → Λe1 depends on a parameter l ∈ k and that every l ∈ k gives such a homomorphism.

3 Let A be a k-algebra, e ∈ A be an idempotent and M be an A-module. a) Show that eAe is a k-algebra and that e is the identity element.

For two A-modules N1, N2, we denote by HomA(N1,N2) the space of A-module morphism from N1 to N2.

b) Show that eM and that HomA(Ae, M) are eAe-modules. Hint: For f ∈ HomA(Ae, M) and a ∈ eAe, the action a·f is given by (a·f)(x) = f(ax) for all x ∈ Ae.

c) Show that the map θM : HomA(Ae, M) → eM, f 7→ f(e) is an isomorphism of eAe-modules.

January 31, 2019 Page 1 of 2 Exercise set 3

Hint: Show that Ae is generated as an A-module by e, and thus that any morphism f : Ae → M depends only on where you send e. With this in mind, show that Im f ⊂ eM. Then show that for any m ∈ eM, there exists a morphism fm : Ae → M such that m ∈ Im fm. Given a ring R, we define the opposite ring Rop by Rop = R as a group, and the multiplication ∗ of Rop is given by a ∗ b = b · a, where · is the multiplication in R. op ∼ d) Show that EndA(Ae) = eAe as k-algebras via the map θAe.

e) Conclude using d) and #2c) that Λei is indecomposable for each i, where Λ is the algebra from problem 2 (In fact it is true for any path algebra Λ and trivial path ei by the same reasonning).

January 31, 2019 Page 2 of 2