On coefficient modules of arbitrary modules

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(Mₖ/M) < ∞ and M⊆ Mₛ⊆⋯⊆ M₁⊆ M₀⊆ q(M), such that (P_Mₖ/M(n)) < s-k, where q ( M ) is the relative integral closure of $M,$ defined by q(M):=M̄∩ Mˢᵃᵗ, where Mˢᵃᵗ=∪n 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.