On finitely generated modules whose first nonzero Fitting ideals are regular
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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 $1leq ileq n$, $jin Lambda$, 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 $M_lambda=R^n/<(a_{1lambda},...,a_{nlambda})^t>$. 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.
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Journal title
volume 8 issue 1
pages 9- 18
publication date 2018-01-01
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