On $v$-domains: a survey

Abstract

An integral domain $D$ is a $v$--domain if, for every finitely generated nonzero (fractional) ideal $F$ of $D$, we have $(F{F}^{-1}{)}^{-1}=D$. The $v$--domains generalize Prüfer and Krull domains and have appeared in the literature with different names. This paper is the result of an effort to put together information on this useful class of integral domains. In this survey, we present old, recent and new characterizations of $v$--domains along with some historical remarks. We also discuss the relationship of $v$--domains with their various specializations and generalizations, giving suitable examples.

40 pages: some typos corrected. To appear in "Commutative Algebra: Noetherian and non-Noetherian Perspectives", Springer, New York