Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

Abstract

In this article, we study the notion of radical perfectness in Prüfer and classical pullbacks issued from valuation domains. We answer positively a question by Erdogdu of whether a domain R such that every prime ideal of the polynomial ring R[X] is radically perfect is one-dimensional. Particularly, we prove that Prüfer and pseudo-valuation domains R over which every prime ideal of the polynomial ring R[X] is radically perfect are one-dimensional domains. Moreover, the class group of such a Prüfer domain is torsion.

Key Words:

• Classical pullback
• Prüfer domain
• Radically perfect
• Quasi-Prüfer domain
• PVD
• UMT-domain

2000 Mathematics Subject Classification:

• Primary 13G055, 13A15, 13F05
• Secondary 13G05, 13F30

ACKNOWLEDGMENT

I would like to thank to the referee for his/her helpful suggestions which helped in improving the quality of this article.

This work is funded by the Deanship of Scientific Research at KFUPM under Project #SB101026.

Notes

Communicated by I. Swanson.