Fitting ideals and cyclic decomposition of finitely generated modules

Abstract EN

Let R be a valuation ring and let M be a finitely generated R-module whose Fitting ideals are principal ideals.

Then M≃⨁ i=1 i=n R/(a i ), where n is the minimal number of generators of M, the a i are in R, and a i divides a i+1 .

If furthermore the Fitting ideals of M are nonzero, then M≃⨁ i=1 i=n R/(F k (M):F k+1 (M)), where F k (M) is the kth Fitting ideal of M.

It is also shown that the Fitting ideals characterize the isomorphism class of M, and allow us to decide whether M is finite free or not.