Exercise 11, § 10

Let A be a commutative ring, B an A-algebra, and M an Artinian B-module. Suppose that the module M is finitely generated.

a) Prove that the ring $B_M$ is left Artinian (use Proposition 6 of VIII, p. 9) and then that the B-module M has finite length.

b) Prove that if M is a simple B-module, then the A-module M is isotypical (deduce from Lemma 1 of VIII, p. 81 that the kernel of the homomorphism $A\rightarrow$ End$_B(M)$ is a maximal ideal).

c) Conclude that M is an A-module of finite length.