If N and P are submodules of M that is an R-module and modules (N intersects P) and (N+P) are finitely generated then show that modules N and P are finitely generated.
Consider the exact sequence
0 --> N n P --> N --> N/(N n P) --> 0
now, by the second isomorphism theorem
(N + P)/P ~ N/(N n P)
(where 'n' denotes intersection)
This is a proof regarding modules and exact sequences.