A Generalized Serre's Condition

TL;DR

This paper introduces a generalized version of Serre's condition for Noetherian rings, expanding the theoretical framework and proving analogous results for the new condition.

Contribution

It defines the generalized Serre's condition $(S_{ ext{ell}}^j)$ and extends existing results to this broader context.

Findings

Established properties of rings satisfying $(S_{ ext{ell}}^j)$

Proved generalizations of classical Serre's condition results

Enhanced understanding of depth and dimension relations in Noetherian rings

Abstract

Throughout, let $R$ be a commutative Noetherian ring. A ring $R$ satisfies Serre's condition $(S_{\ell})$ if for all $P \in \Spec R,$ $\depth R_P \geq \min \{ \ell , \dim R_P \}$. Serre's condition has been a topic of expanding interest. In this paper, we examine a generalization of Serre's condition $(S_{\ell}^j)$. We say a ring satisfies $(S_{\ell}^j)$ when $\depth R_P \geq \min \{ \ell , \dim R_P -j \}$ for all $P \in \Spec R$. We prove generalizations of results for rings satisfying Serre's condition.