A note on $UFD$

TL;DR

This paper investigates conditions under which prime ideals in certain local rings are principal, providing simplified proofs of known theorems and new insights into the structure of rings with low multiplicity.

Contribution

It establishes principal ideal conditions in specific local rings and offers simplified proofs of classical theorems, advancing understanding of ring structure and ideal properties.

Findings

Prime ideals of height one are principal under certain depth conditions.

Reproves the Auslander-Buchsbaum theorem using local cohomology.

Shows rings of multiplicity at most three are hypersurfaces.

Abstract

We search for principal ideals. As a sample, let $R$ be a strongly-normal, almost-factorial, and complete-intersection local ring with a prime ideal $P$ of height one. If $depth(R/ P)\geq dim R-2$, we show $P$ is principal. As an immediate corollary, we apply some easy local cohomology arguments and reprove a celebrated theorem of Auslander-Buchsbaum, simplifying a result of Dao and Samuel. From this, we show the hypersurface property of rings of multiplicity at most three. As another application, we answer affirmatively a question posted by Braun.