On coefficient modules of arbitrary modules

Abstract 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 [10], we show the existence of a unique largest R -module M k with $\ell_R(M_{k}/M) \lt \infty$ and $M\subseteq M_{s}\subseteq\cdots\subseteq M_{1}\subseteq M_{0}\subseteq q(M),$ such that $\deg(P_{M_{k}/M}(n)) \lt s-k,$ where q ( M ) is the relative integral closure of $M,$ defined by $q(M):=\overline{M}\cap M^{sat},$ where $M^{sat}=\cup_{n\geqslant 1}(M:_F\mathfrak{m}^n)$ 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 0th Fitting ideal of $F/M$ , and discuss their structural properties. Finally, we present some applications and discuss some properties.