Exercise 10, § 12

Keep the assumptions of Exercise 9 and suppose, moreover, that the B-module N is simple; denote its commutant by D.

a) The mapping $\vartheta$ is surjective if and only if M admits a finite basis over A or D is finite-dimensional over K (use Exercise 9, c)).

b) Prove that the $(A\otimes$ B)-module $M\otimes N$ is semisimple if and only if the ring $A\otimes D^o$ is semisimple (reduce to the case $M = A_s$; observe that $M\otimes N$ is a free $A\otimes D^o$-module, and use Proposition 5 of VIII, p. 85 as well as Exercise 9, a)).