Some new invariants of Noetherian local rings related to square of the maximal ideal

TL;DR

This paper introduces two new invariants for Noetherian local rings related to the generators of reductions of the maximal ideal, analyzing their properties and computing them for specific classes of rings, including quadratic quotients and graph edge ideals.

Contribution

It defines and studies two novel invariants measuring generators of reductions of the maximal ideal in Noetherian local rings, with explicit computations for quadratic and graph-related quotients.

Findings

Computed invariants for quadratic ideal quotients of polynomial rings.

Analyzed invariants in the context of edge ideals of graphs.

Provided properties and bounds for these invariants.

Abstract

We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider the minimum among the number of generators of ideals $I$ such that either $I^2 = \mathfrak m^2$ or $I \supseteq \mathfrak m^2$ holds. We investigate subsequently the case that $R$ is the quotient of a polynomial ring $k[x_1, \dots, x_n]$ by an ideal $I$ generated by homogeneous quadratic forms, and we compute these invariants. We devote specific attention to the case that $R$ is the quotient of a polynomial ring $k[x_1, \dots, x_n]$ by the edge ideal of a finite simple graph $G.$