Edge Ideals of Prime Ideal Graphs: Ordinary Powers, Polymatroidality, and Analytic Spread

TL;DR

This paper characterizes the structure of prime ideal graphs in finite rings, describes their edge ideals' powers, and explores algebraic properties like polymatroidality and analytic spread.

Contribution

It provides a complete description of prime ideal graphs, their edge ideals, and the algebraic properties of their powers, including explicit formulas and resolutions.

Findings

Prime ideal graphs are complete split graphs with a specific structure.

The minimal generators of the powers of the edge ideal are characterized explicitly.

All ordinary powers are polymatroidal with linear quotients and linear resolutions.

Abstract

Let $R$ be a finite commutative ring with identity, and let $P$ be a proper prime ideal of $R$. The prime ideal graph $\Gamma_P(R)$ has vertex set of $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy\in P$. We prove that $\Gamma_P(R)\cong K_{|P|-1}\vee \overline{K}_{|R|-|P|}$, so prime ideal graphs form a ring-induced family of complete split graphs. Using this description, we determine the minimal vertex covers and obtain an irredundant primary decomposition of the edge ideal $I(\Gamma_P(R))$. For every $n\geq 1$, we characterize the minimal monomial generators of the ordinary power $I(\Gamma_P(R))^n$: a monomial $x^\alpha y^\beta$ belongs to $G(I(\Gamma_P(R))^n)$ if and only if $|\alpha|+|\beta|=2n, \ |\beta|\leq n$, and $0\leq \alpha_i\leq n$ for all $i$. Consequently, we derive a closed formula for $\mu(I(\Gamma_P(R))^n)$. We also prove that every ordinary power is polymatroidal and hence has linear quotients and a $2n-$linear resolution. Finally, we interpret $\mu(I(\Gamma_P(R))^n)$ as the Hilbert function of the special fiber ring and compute the analytic spread of $I(\Gamma_P(R))$.