# On the diameter of an ideal

## TL;DR

This paper introduces the concept of the diameter of an ideal in a polynomial ring, measuring the distance between minimal primes, and identifies classes of ideals with bounded diameter.

## Contribution

It defines the diameter of an ideal and characterizes large classes of Hirsch ideals with bounded diameter, including quadratic radical and Gorenstein ideals.

## Key findings

- Quadratic radical ideals of codimension ≤ 4 have bounded diameter.
- Ideals with a square-free complete intersection initial ideal are Hirsch ideals.
- The diameter measures the distance between minimal primes of an ideal.

## Abstract

We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals with diameter not larger than the codimension, such as: quadratic radical ideals of codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a square-free complete intersection initial ideal.