t-Local Domains and 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. Dedicated to David F. Andersonshorter, more efficient proof of Theorem 6.2 of [27]. We also point to examples of maximal t-ideals Q in a particular domain D such that D Q is not t-local. Background results and t-local domainsWe start with proving some important preliminary results. But, for that, we need to recall the formal definition of star operation. A star operation on D is a map

Domains whose ideals meet a universal restriction

“…Indeed there do exists examples of Noetherian domains with maximal t-ideals of height greater than one, see e.g. [22,Example 3.5].…”