On finitely generated modules whose first nonzero Fitting ideals are regular

A finitely generated R-module is said to be a module of type (F r) 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=(a ij ) with a ij ∈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 (F n−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 M λ =R n /. 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.