Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-58538120180101On finitely generated modules whose first nonzero Fitting ideals are regular91833815ENSomayeh HadjirezaeiDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.Somayeh KarimzadehDepartment of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.Journal Article20160508A 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 $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set. Let $M=R^n/N$ be a module of type ($F_{n-1}$) 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 $ lambdain Lambda $, put $M_lambda=R^n/$. The main result of this paper asserts that if $M_lambda $ is a regular $R$-module, for some $lambdainLambda$, 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 $lambdain Lambda$, then $ Mcong M_lambda. $ As a consequence we characterize all non-torsionfree modules over a regular ring, whose first nonzero Fitting ideals are maximal.http://cgasa.sbu.ac.ir/article_33815_eb94849dbfc998e1f81615c7347eb37f.pdf