The $\mathfrak{a}$-Filter grade of an ideal $\mathfrak{b}$ and $(\mathfrak{a},\mathfrak{b})$-$\mathrm{f}$-modules

TL;DR

This paper investigates the $ extrm{f} extrm{-} extrm{grad}$ of an ideal pair in a Noetherian ring, providing computations, bounds, and structural properties of related modules, extending the understanding of Cohen-Macaulay-like modules.

Contribution

It introduces new bounds and structural insights for $(rak{a},rak{b})$-$ extrm{f}$-modules, extending previous work on $ extrm{f} extrm{-} extrm{grad}$ in commutative algebra.

Findings

Computed bounds for $ extrm{-} extrm{grad}_R(rak{a},rak{b},M)$

Described the structure of $(rak{a},rak{b})$-$ extrm{-}f$-modules

Identified properties analogous to Cohen-Macaulay modules

Abstract

Let $\mathfrak{a},\mathfrak{b}$ be two ideals of a commutative noetherian ring $R$ and $M$ a finitely generated $R$-module.~We continue to study $\textrm{f}\textrm{-}\mathrm{grad}_R(\mathfrak{a},\mathfrak{b},M)$ which was introduced in [Bull. Malays. Math. Sci. Soc. 38 (2015) 467--482], some computations and bounds of $\textrm{f}\textrm{-}\mathrm{grad}_R(\mathfrak{a},\mathfrak{b},M)$ are provided.~We also give the structure of $(\mathfrak{a},\mathfrak{b})$-$\mathrm{f}$-modules,~various properties which are analogous to those of Cohen Macaulay modules are discovered.