A Characterization of locally quasi-unmixed rings

TL;DR

This paper characterizes locally quasi-unmixed rings by the equivalence of certain ideal topologies and explores how these topologies behave under ring homomorphisms.

Contribution

It provides a new characterization of locally quasi-unmixed rings using the equivalence of topologies defined by integral closures and symbolic powers of ideals.

Findings

Topologies defined by integral closures and symbolic powers are equivalent if and only if the ring is locally quasi-unmixed.

Results on the behavior of these topologies under various ring homomorphisms.

Establishes a link between ring properties and ideal topologies.

Abstract

Let $\bar{I}$ denote the integral closure of an ideal in a Noetherian ring $R$. The main result of this paper asserts that $R$ is locally quasi-unmixed if and only if, the topologies defined by $\overline{I^n}$ and $I^{\langle n\rangle}$, $\ n\geq 1$, are equivalent. In addition, some results about the behavior of linearly equivalent topologies of ideals under various ring homomorphisms are included.