Bounds for numbers of generators for a class of submodules of a finitely generated module

The aim of this paper is to obtain a uniform bound for a certain class of submodules from the following theorem: Let $(R,\mathfrak{m})$ be a local ring, let $M$ be a finite $R$–module of dimension $d\ge 1$ and let $\mathfrak{q}$ be an ideal of $R$ generated by a system of parameters on $M$. Let $N$ be a submodule of $M$ with $\text{depth}M\mathrm{/}N\ge d-1$. Then $\mathrm{\ell }(N\mathrm{/}\mathfrak{q}N)\le \mathrm{\ell }(M\mathrm{/}\mathfrak{q}M)$.