# The finiteness dimension of modules and relative Cohen-Macaulayness

## TL;DR

This paper investigates the finiteness dimension and adjusted depth of modules relative to ideals in a Noetherian ring, focusing on modules that are Cohen-Macaulay relative to a specific ideal, and explores their applications.

## Contribution

It introduces and analyzes the concepts of finiteness dimension and adjusted depth for modules relative to ideals, especially in the context of relative Cohen-Macaulay modules, providing new insights and applications.

## Key findings

- Characterization of finiteness dimension for relative Cohen-Macaulay modules.
- Relationships between adjusted depth and finiteness dimension.
- Applications to module theory and ideal-related properties.

## Abstract

Let $R$ be a commutative Noetherian ring, $\mathfrak a$ and $\mathfrak b$ ideals of $R$. In this paper, we study the finiteness dimension $f_{\mathfrak a}(M)$ of $M$ relative to $\mathfrak a$ and the $\mathfrak b$-minimum $\mathfrak a$-adjusted depth $\lambda_{\mathfrak a}^{\mathfrak b}(M)$ of $M$, where the underlying module $M$ is relative Cohen-Macaulay w.r.t $\mathfrak a$. Some applications of such modules are given.