An Abstract Algebra Question on the structure theorem

Question

Find an example of an integral domain R (but make sure it isn't a PID) and a module M that is finitely generated but doesn't satisfy the conclusion of The Fundamental Theorem of Finitely Generated Modules Over a PID (in other words the structure theorem). Justification for M should be included.

Answer 1

Consider M = Z[x]/(2,x) as a module over R = Z[x]. Since Z[x]/(2,x) is isomorphic to Z/2Z as rings, M has only two elements; in particular, M is a finitely generated R-module. Since Z[x]/(f(x)) is infinite for all polynomials f(x) in Z[x], and since M is finite, the structure theorem cannot hold for M.