Prüfer conditions in the Nagata ring and the Serre’s conjecture ring

ABSTRACT

The Nagata ring R(X) and the Serre’s conjecture ring R⟨X⟩ are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this paper, we contribute to the study of Prüfer conditions in R(X) and R⟨X⟩. In particular, we solve the four open questions posed by Glaz and Schwarz in Section 8 of their survey paper [38] related to the transfer of Prüfer conditions to these two constructions.

KEYWORDS:

• Arithmetical ring
• fqp-ring
• Gaussian ring
• Nagata ring
• Prüfer domain
• Prüfer ring
• semihereditary ring
• Serre’s conjecture ring
• weak dimension

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F20
• 13B25
• 13C10
• 13F30
• 13D05