Hadjirezaei, S., Karimzadeh, S. (2018). On finitely generated modules whose first nonzero Fitting ideals are regular. Categories and General Algebraic Structures with Applications, 8(1), 9-18.

Somayeh Hadjirezaei; Somayeh Karimzadeh. "On finitely generated modules whose first nonzero Fitting ideals are regular". Categories and General Algebraic Structures with Applications, 8, 1, 2018, 9-18.

Hadjirezaei, S., Karimzadeh, S. (2018). 'On finitely generated modules whose first nonzero Fitting ideals are regular', Categories and General Algebraic Structures with Applications, 8(1), pp. 9-18.

Hadjirezaei, S., Karimzadeh, S. On finitely generated modules whose first nonzero Fitting ideals are regular. Categories and General Algebraic Structures with Applications, 2018; 8(1): 9-18.

On finitely generated modules whose first nonzero Fitting ideals are regular

^{}Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O. Box 7718897111, Rafsanjan, Iran.

Abstract

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.