Exercise 7, § 4

Let A be a ring and M an A-module. A submodule N of M is called essential if $N\cap P\not= 0$ for every nonzero submodule P of M.

a) Prove that every submodule N of M is a direct factor of an essential submodule (consider a submodule P that is maximal among the submodules of A with zero intersection with N).

b) Prove that the socle of M is the intersection of the essential submodules of M (deduce from a) that this intersection is semisimple). In particular, M is semisimple if and only if it does not contain any essential submodules other than itself.