On super v-domains

Abstract

An integral domain D, with quotient field K, is a v-domain if for each nonzero finitely generated ideal A of D we have ${(A{A}^{-1})}^{-1}=D.$ It is well known that if D is a v-domain, then some quotient ring D_S of D may not be a v-domain. Calling D a super v-domain if every quotient ring of D is a v-domain we characterize super v-domains as locally v-domains. Using techniques from factorization theory we show that D is a super v-domain if and only if $D[X]$ is a super v-domain if and only if $D+XK[X]$ is a super v-domain and give new examples of super v-domains that are strictly between v-domains and P-domains, domains that are essential along with all their quotient rings.

Keywords:

• Locally v-domain
• P-domain
• splitting set
• super v-domain
• t-splitting set

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13F05
• 13G05
• Secondary: 13B25
• 13B30