Characterizing Regular Local Rings via Analogues of (*)-properties

TL;DR

This paper introduces new properties and relations in commutative algebra to characterize regular local rings, refining previous criteria and expanding their applications.

Contribution

It develops $(A)$-properties, $(B)$-properties, and $(C)$-relations as analogues of existing properties, providing a refined criterion for regularity in local rings.

Findings

Established a new criterion for regular local rings.

Refined previous results by Ghosh, Gupta, and Puthenpurakal.

Extended applications of the new properties and relations.

Abstract

Let $R$ be a commutative noetherian local ring. As analogues of $(*)$-properties introduced by Ghosh, Gupta, and Puthenpurakal, we introduce and study $(\mathrm{A})$-properties, $(\mathrm{B})$-properties, and $(\mathrm{C})$-relations. Using these three notions, we establish a criterion for a local ring to be regular. This recovers and refines the main result of Ghosh, Gupta, and Puthenpurakal and has further applications.