(a) Let M be an R-module over a PID R such that M(p) 6= 0 for infinitely many (non-associate) primes. Show that M is not finitely generated. (b) Let R be a PID and M a f.g. torsion R-module such that there exists m M, ann(m) = (r), where r = order(M). What can you say about invariant factors of M?