Let $R$ be a commutative ring and let $N$ be a submodule of $R^n$ which consists of columns of a matrix $A=(a_{ij})$ with $a_{ij}\in R$ for all $1\le i\le n$, $j\in \mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is an index set. For every $\mu =\{j_1,\dots ,j_q\}\subseteq \mathrm{\Lambda }$, let I${}_{\mu }(N)$ be the ideal generated by subdeterminants of size $q$ of the matrix $(a_{ij}:1\le i\le n,j\in \mu )$. Let $M=R^n/N$. In this paper, we obtain a constructive description of T($M$) and we show that when $R$ is a local ring, $M$/T($M$) is free of rank $n-q$ if and only if I${}_{\mu }(N)$ is a principal regular ideal, for some $\mu =\{j_1,\dots ,j_q\}\subseteq \mathrm{\Lambda }$. This improves a lemma of Lipman which asserts that, if I($M$) is the $(m-q)$th Fitting ideal of $M$ then I($M$) is a regular principal ideal if and only if $N$ is finitely generated free and $M$/T($M$) is free of rank $m-q.$

让$R$是一个交换环并且让$ñ$成为的子模块$R^n$它由矩阵的列组成$\mathrm{一个}=({\mathrm{一个}}_{\mathrm{一世}j})$和${\mathrm{一个}}_{\mathrm{一世}j}\in R$对所有人$1\le \mathrm{一世}\le n$,$j\in \mathrm{\Lambda }$， 在哪里$\mathrm{\Lambda }$是一个索引集。对于每一个$\mu =\{j_1,\dots ,j_q\}\subseteq \mathrm{\Lambda }$，让我${}_{\mu }(ñ)$是由大小的子行列式生成的理想$q$矩阵的$({\mathrm{一个}}_{\mathrm{一世}j}：1\le \mathrm{一世}\le n,j\in \mu )$. 让$米=R^n/ñ$. 在本文中，我们获得了 T($米$) 我们证明了当$R$是本地环，$米$/T($米$) 是无等级的$n-q$当且仅当我${}_{\mu }(ñ)$是一个主要的常规理想，对于一些$\mu =\{j_1,\dots ,j_q\}\subseteq \mathrm{\Lambda }$. 这改进了 Lipman 的一个引理，它断言，如果 I($米$） 是个$(米-q)$的拟合理想$米$然后我（$米$) 是正则主理想当且仅当$ñ$是有限生成的自由和$米$/T($米$) 是无等级的$米-q.$