  Let $(R, \mathfrak{m})$ be a $d$-dimensional Noetherian local ring that is
formally equidimensional, and let $M$ be an arbitrary $R$-submodule of the free
module $F = R^p$ with an analytic spread $s:=s(M)$. In this work, inspired by
Herzog-Puthenpurakal-Verma in \cite{herzog}, we show the existence of an unique
largest $R$-module $M_{k}$ with $\ell_R(M_{k}/M)<\infty$ and $M\subseteq
M_{s}\subseteq\cdots\subseteq M_{1}\subseteq M_{0}\subseteq q(M),$ such that
$\deg(P_{M_{k}/M}(n))<s-k,$ where $q(M)$ is the relative integral closure of
$M,$ defined by $q(M):=\overline{M}\cap M^{sat},$ where $M^{sat}=\cup_{n\geq
1}(M:_F\mathfrak{m}^n)$ is the saturation of $M$. We also provide a structure
theorem for these modules. Furthermore, we establish the existence of
coefficient modules between $I(M)M$ and $M$, where $I(M)$ denotes the $0$-th
Fitting ideal of $F/M$, and discuss their structural properties. Finally, we
present some applications and discuss some properties.
