$\star$-super potent domains

Abstract

For a finite-type star operation ⋆ on a domain R, we say that R is ⋆-super potent if each maximal ⋆-ideal of R contains a finitely generated ideal I such that (1) I is contained in no other maximal ⋆-ideal of R and (2) J is ⋆-invertible for every finitely generated ideal J⊇I. Examples of t-super potent domains include domains each of whose maximal t-ideals is t-invertible (e.g., Krull domains). We show that if the domain R is ⋆-super potent for some finite-type star operation ⋆, then R is t-super potent, we study t-super potency in polynomial rings and pullbacks, and we prove that a domain R is a generalized Krull domain if and only if it is t-super potent and has t-dimension one.