On super $v$-domains

An integral domain $D,$ with quotient field $K,$ is a $v$-domain if for each nonzero finitely generated ideal $A$ of $D$ we have $(AA^{-1})^{-1}=D.$ 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 that were studied in [Manuscripta Math. 35(1981)1-26]