Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 8 1 2018 01 01 On finitely generated modules whose first nonzero Fitting ideals are regular 9 18 33815 10.29252/cgasa.8.1.9 EN Somayeh Hadjirezaei Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran. Somayeh Karimzadeh Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran. Journal Article 2016 05 08 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 $a_{ij}\in R$ for all $1\leq i\leq n$, $j\in \Lambda$, where $\Lambda $ is a (possibly infinite) index set.  Let  $M=R^n/N$ be  a module of type ($F_{n-1}$) 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 $ \lambda\in \Lambda $, put $M_\lambda=R^n/<(a_{1\lambda},...,a_{n\lambda})^t>$. 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. https://cgasa.sbu.ac.ir/article_33815_eb94849dbfc998e1f81615c7347eb37f.pdf