\(t\)-local domains and valuation domains (Q2331225)

An integral domain is called \(t\)-local if it is local and its maximal ideal is a \(t\)-ideal. An example of a \(t\)-local domain is a valuation domain \((V, M)\). This is because every nonzero finitely generated ideal \(J\) of \(V\) is principal and so, \(J = J^{t}\); hence the maximal ideal \(M\) is a \(t\)-ideal. However, a localization of a \(t\)-local domain is not necessary \(t\)-local (see Example 2.9). This yields the following question: under what conditions is a \(t\)-local domain a valuation domain? The main purpose of this paper is to address this question. Properties of \(t\)-local domains are studies. Examples of \(t\)-local domains are given, and conditions under which a ring is a \(t\)-local domain are studies. For the entire collection see [Zbl 1412.13002].