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-۱)-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}\in R for all ۱\leq i\leq n, j\in \Lambda, where \Lambda is a (possibly infinite) index set.  Let  M=R^n/N be  a module of type (F_{n-۱}) and {\rm T}(M) be the submodule of M consisting of all elements of M that are annihilated by a regular element of R. For \lambda\in \Lambda , put M_\lambda=R^n/<(a_{۱\lambda},...,a_{n\lambda})^t>. The main result of this paper asserts that if M_\lambda is a regular R-module, for some \lambda\in\Lambda, then M/{\rm T}(M)\cong M_\lambda/{\rm T}(M_\lambda). Also it is shown that if M_\lambda is a regular torsionfree R-module, for some \lambda\in \Lambda, then M\cong M_\lambda. As a consequence we characterize all  non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.