Exercise 9, § 12

Let A and B be K-algebras, M a free A-module, and N a B-module.

a) Prove that the image of the canonical homomorphism $\vartheta$ from End$_A(M)\otimes$End$_B(N)$ to End$_{A\otimes B}(M\otimes N)$ is a dense subring of the ring End$_{A\otimes B}(M\otimes N)$.

b) If the ring of homotheties of the B-module N is dense in its bicommutant, then the same holds for the ring of homotheties of the $(A\otimes$ B)-module $M\otimes N$.

c) Suppose that there exist an element $y$ of N and a sequence $(v_n)$ of endomorphisms of N such that the elements $v_n(y)$ generate an infinite-dimensional vector space over K. Prove that the mapping $\vartheta$ is not surjective.