On some classes of integral domains with finitely many star operations of finite type
In this article, the authors investigate classes of integral domains in which there are finitely many star operations of finite type. Let \(R\) be an integral domain with quotient field \(L\), \(F(R)\) the set of nonzero fractional ideals of \(R\) and \(f(R)\) the set of nonzero finitely generated fractional ideals of \(R\). A mapping \(* : F(R) \to F(R)\), \(E \to E^*\), is called a star operation on \(R\) if the following conditions hold for all \(a \in L \setminus \{0\}\) and \(E\), \(F \in F(R)\): (1) \((aE)^* = aE^*\), (2) \(E \subseteq E^*\) and if \(E \subseteq F\) then \(E^* \subseteq F^*\), and (3) \((E^*)^*=E^*\). A star operation \(*\) is said to be of finite type (or of finite character) if for each nonzero (fractional) ideal \(E\) of \(R\), \(E^* = \cup F^*\) where the union is taken over all nonzero finitely generated subideals \(F\) of \(E\). The authors begin by investigating integral domains with at most two star operations of finite type where they are able to generalize several analogous results. Then the authors move to study Mori domains with only finitely many star operations of finite type wheer they aim to generalize certain results in the Noetherian case. The final section studies pullbacks. The paper includes several nice examples along the way to help illustrate much of the theory presented in the article. In addition to an extensive bibliography, there are several open questions/conjectures presented in the article that may be of interest.
scientific article; zbMATH DE number 7471603
scientific article