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.