I know that one can compute Fitting ideals of a finitely presented module (over a commutative ring with identity). However, are they the only invariants of such a module?
In other words, my question is: if two finitely presented modules have the same Fitting ideals, then are they isomorphic? Obviously the answer is yes if the base ring is a PID. If the answer is no, how is a strategy to prove that two f.p. modules with the same Fitting ideals are not isomorphic?
Unfortunately, base change is not useful in this problem.