On Coefficient Module of Arbitrary Modules

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.