$\star $-super potent domains

TL;DR

This paper introduces the concept of $ ext{super potent}$ domains related to star operations, explores their properties, and characterizes generalized Krull domains via $t$-super potency and $t$-dimension.

Contribution

It defines $ ext{super potent}$ domains for star operations, establishes their relation to $t$-super potency, and characterizes generalized Krull domains through these properties.

Findings

$ ext{Super potent}$ domains include domains with $t$-invertible maximal $t$-ideals.

If a domain is $ ext{super potent}$ for some star operation, it is $t$-super potent.

A domain is generalized Krull iff it is $t$-super potent with $t$-dimension one.

Abstract

For a finite-type star operation $\star$ on a domain $R$, we say that $R$ is $\star$-super potent if each maximal $\star$-ideal of $R$ contains a finitely generated ideal $I$ such that (1) $I$ is contained in no other maximal $\star$-ideal of $R$ and (2) $J$ is $\star$-invertible for every finitely generated ideal $J \supseteq I$. 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 $\star$-super potent for some finite-type star operation $\star$, 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.