Exercise 1, § 4

a) Let M be an A-module and $x$ an element of M whose annihilator is the intersection of a finite family of maximal ideals. Prove that the module $Ax$ is semisimple (embed it in a finite product of simple modules).

b) Deduce that a module M is semisimple if and only if the annihilator of every nonzero element is the intersection of a finite family of maximal ideals.