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 R n 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 = R n /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

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... of R. For λ ∈ Λ, put M λ = R n / < (a 1λ , ..., a nλ) t >. The main result of this paper asserts that if M λ is a regular R-module, for some λ ∈ Λ, then M/T(M) ∼ = M λ /T(M λ). Also it is shown that if M λ is a regular torsionfree R-module, for some λ ∈ Λ, then M ∼ = M λ. As a consequence we characterize all non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.