# Locally unmixed modules and linearly equivalent topologies

## TL;DR

This paper characterizes locally unmixed modules over Noetherian rings through the linear equivalence of topologies defined by symbolic and adic powers of certain ideals, linking algebraic properties to topological behavior.

## Contribution

It establishes a new equivalence between local unmixedness of modules and the linear equivalence of specific topologies, extending understanding of module and ideal interactions.

## Key findings

- Locally unmixed modules are characterized by topology equivalence.
- The topology defined by symbolic powers aligns with the $I$-adic topology under certain conditions.
- The result provides a new criterion for local unmixedness in terms of topological properties.

## Abstract

Let $R$ be a commutative Noetherian ring, and let $N$ be a non-zero finitely generated $R$-module. The purpose of this paper is to show that $N$ is locally unmixed if and only if, for any $N$-proper ideal $I$ of $R$ generated by $\Ht_N I$ elements, the topology defined by $(IN)^{(n)}$, $n \geq 0$, is linearly equivalent to the $I$-adic topology.