Exercise 9, § 5

Let D be a field and V a right vector space over D. We view V as a left module over the ring A = End$_D(V)$.

a) Prove that the A-module V is projective, finitely generated, faithful, and balanced. Prove that the $(D$, A)-bimodules Hom$_A(V,A_s)$ and $V^*=$ Hom$_D(V,D_d)$ are canonically isomorphic.

b) Prove that the A-module V is generating if and only if the dimension of the D-vector space V is finite.

c) What is the trace ideal of the A-module V?