Volume 8, Number 1, January 2018, 9-18. On finitely generated modules whose first nonzero Fitting ideals are regular Somayeh Hadjirezaei ∗ and Somayeh Karimzadeh Abstract. 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 element 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. * Corresponding author Keywords : Fitting ideals, type of a module, torsion submodule. Mathematics Subject Classification [2010]: 13C05, 13C10, 13C12. Received: 8 May 2016, Accepted: 15 September 2016 ISSN Print: 2345-5853 Online: 2345-5861 © Shahid Beheshti University 9