# $t$-local domains and valuation domains

## TL;DR

This paper investigates the conditions under which $t$-local domains are valuation domains, contrasting their properties and localizations with those of valuation domains, and surveys existing related work.

## Contribution

It provides a comprehensive analysis of when $t$-local domains can be characterized as valuation domains, extending previous research and clarifying their relationship.

## Key findings

- Localization of a $t$-local domain need not be $t$-local
- Valuation domains are always $t$-local and their localizations remain valuation domains
- Conditions are identified under which $t$-local domains are valuation domains

## Abstract

In a valuation domain $(V,M)$ every nonzero finitely generated ideal $J$ is principal and so, in particular, $J=J^t$, hence the maximal ideal $M$ is a $t$-ideal. Therefore, the $t$-local domains (i.e., the local domains, with maximal ideal being a $t$-ideal) are "cousins" of valuation domains, but, as we will see in detail, not so close. Indeed, for instance, a localization of a $t$-local domain is not necessarily $t$-local, but of course a localization of a valuation domain is a valuation domain. So it is natural to ask under what conditions is a $t$-local domain a valuation domain? The main purpose of the present paper is to address this question, surveying in part previous work by various authors containing useful properties for applying them to our goal.