Prüfer rings in a certain pullback

Abstract

Let D be an integral domain with quotient field K, X be an indeterminate over D, $K[X]$ be the polynomial ring over K, $n\ge 2$ be an integer, $K[\theta ]=K[X]/(X^n)$, where $\theta =X+(X^n)$, and $R_n=D+\theta K[\theta ]$, i.e, $R_n=\{f+(X^n)\in K[X]/(X^n)|f(0)\in D\}$. Then R_n is a subring of $K[\theta ]$ with total quotient ring $K[\theta ]$. In this paper, we study several ring-theoretic properties of R_n, with a focus on Prüfer rings and Prüfer v-multiplication rings (PvMRs). For example, we show when R_n is a Prüfer ring, a Bézout ring, a PvMR, or a GCD ring.

KEYWORDS:

• $D+\theta K[\theta ]$ construction
• Prüfer ring
• Bézout ring
• PvMR
• GCD ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13F05

Acknowledgment

This work was supported by the Incheon National University Research Grant in 2022. The authors would like to thank the reviewer for his/her constructive feedback, which significantly improved the manuscript.