On the first nonzero Fitting ideal of a module

Let R be a commutative ring and K be a submodule of Rm, and let I be the first nonzero Fitting ideal of the module M=Rm/K. A lemma of Lipman asserts that if R is quasilocal and I is the (m−q)th Fitting ideal of M, then I is regular principal if and only if K is finitely generated free and M/T(M) is free of rank m−q. (Here T(M) is the submodule of M consisting of all elements of M that are annihilated by a regular element of R.) This paper contains two global generalizations of this result, one with the hypothesis that I is principal regular and the other with the hypothesis that I is invertible.