On $v$-domains: a survey

Details On $v$-domains: a survey On $v$-domains: a survey

2009-02-20

arxiv.org/abs/0902.3592v2

Mathematics

Commutative Algebra

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

Scientific paper

An integral domain $D$ is a $v$--domain if, for every finitely generated nonzero (fractional) ideal $F$ of $D$, we have $(FF^{-1})^{-1}=D$. The $v$--domains generalize Pr\"{u}fer 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.

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