On coefficient modules of arbitrary modules

Abstract Let (R, 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 = Rᵖ 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 R (M₊/M) and M Mₒ M₁ M₀ q (M), such that (P₌_₊/M (n) ) s-k, where q (M) is the relative integral closure of M, defined by q (M): =M M^sat, where M^sat=₍ ₁ (M: Fmⁿ) 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.