Uppers to Zero in Polynomial Rings and Prüfer-Like Domains

Abstract

Let D be an integral domain and X an indeterminate over D. It is well known that (a) D is quasi-Prüfer (i.e., its integral closure is a Prüfer domain) if and only if each upper to zero Q in D[X] contains a polynomial g ∈ D[X] with content c _D (g) = D; (b) an upper to zero Q in D[X] is a maximal t-ideal if and only if Q contains a nonzero polynomial g ∈ D[X] with c _D (g) ^v  = D. Using these facts, the notions of UMt-domain (i.e., an integral domain such that each upper to zero is a maximal t-ideal) and quasi-Prüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this article, given a semistar operation ☆ in the sense of Okabe–Matsuda, we introduce the ☆-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UMt-domains and the Prüfer v-multiplication domains.

Key Words:

• Prüfer domain
• Prüfer v-multiplication domain
• Quasi-Prüfer domain
• Star and semistar operation
• UMt-domain
• Upper to zero

2000 Mathematics Subject Classification:

• 13F05
• 13A15
• 13G05
• 13B25

ACKNOWLEDGMENTS

During the preparation of this paper, the second named author was partially supported by a grant PRIN-MiUR.

Notes

²Added in Proofs: This problem was solved by the authors in case of a stable semistar operation of finite type. The corresponding article “Uppers to zero and semistar operations in polynomial rings” is now published in Journal of Algebra 318:484–493 (2007).

Communicated by R. H. Villarreal.