Finitely presented modules

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.

The Fitting ideals of a finitely presented module $M$ over a ring $A$ do no not determine the module.
Indeed suppose the ring $A$ is connected (no idempotents $\neq0,1$).
Then $all$ finitely presented projective modules $M$ of rank $r$ have the same sequence of Fitting modules, namely
$$F^0(M)=F^1(M)=... =F^{r-1} (M)=0\subsetneq F^r(M)=F^{r+1}(M)=...=A$$
• @Georges, thank you a lot! My question was quite stupid, sorry. To Pierre, take a non principal ideal in a Dedekind domain: for example, $A = \mathbb{Z}[\sqrt{-5}]$ and $I = (2, 1 + \sqrt{-5})$. Commented Jan 7, 2012 at 14:53