On finitely generated modules whose first nonzero Fitting ideals are regular

A finitely generated R-module is said to be a module of type (Fr) if its (r−1)-th Fitting ideal is the zero ideal and its r-th Fitting ideal is a regular ideal. Let R be a commutative ring and N be a submodule of Rn which is generated by columns of a matrix A=(aij) with aij∈R for all 1≤i≤n, j∈Λ, where Λ is a (possibly infinite) index set. Let M=Rn/N be a module of type (Fn−1) and T(M) be the submodule of M consisting of all elements of M that are annihilated by a regular element of R. For λ∈Λ, put Mλ=Rn/ (a1λ,...,anλ)t . The main result of this paper asserts that if Mλ is a regular R-module, for some λ∈Λ, then M/T(M)≅Mλ/T(Mλ). Also it is shown that if Mλ is a regular torsionfree R-module, for some λ∈Λ, then M≅Mλ. As a consequence we characterize all non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.