On t-reductions of ideals in pullbacks

TL;DR

This paper explores t-reductions of ideals within pullback constructions, establishing key properties and transfer results that expand understanding of ideal behavior in various domain classes.

Contribution

It proves the equivalence of finite t-basic and v-basic ideal properties in any domain and extends transfer results of the finite t-basic property to pullback domains.

Findings

Finite t-basic and v-basic ideal properties coincide in any domain.

Transfer of finite t-basic property to pullbacks is established.

New domain classes with finite t-basic property are identified between v-domains and integrally closed domains.

Abstract

This paper investigates t-reductions of ideals in pullback constructions. Section 2 examines the correlation between the notions of reduction and t-reduction in pseudo-valuation domains. Section 3 solves an open problem on whether the finite t-basic and v-basic ideal properties are distinct. We prove that these two notions coincide in any arbitrary domain. Section 4 features the main result, which establishes the transfer of the finite t-basic ideal property to pullbacks in line with Fontana-Gabelli's result on PvMDs and Gabelli-Houston's result on v-domains. This allows us to enrich the literature with new families of examples, which put the class of domains subject to the finite t-basic ideal property strictly between the two classes of v-domains and integrally closed domains.