Edge Ideals of Prime Ideal Graphs over Finite Rings: Ordinary Powers, Fiber Cones, and Linear Powers

Abstract

Let $R$ be a finite commutative ring with identity and let $P$ be a proper prime ideal of $R$. The prime ideal graph $Γ_P(R)$ has vertex set $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy\in P$. We prove that prime ideal graphs form a ring-realizable subfamily of complete split graphs. More precisely, if $m=|P|$, $q=|R/P|$, then $q$ is a prime power and $Γ_P(R)\cong K_{m-1}\vee \overline{K}_{m(q-1)}$. We also prove a realization theorem showing that every complete split graph of this form arises from a prime ideal of a finite commutative ring. For the edge ideal $I=I(Γ_P(R))$, we determine the minimal vertex covers and obtain the irredundant primary decomposition. We characterize the minimal monomial generators of every ordinary power $I^n$ and derive a closed formula for $μ(I^n)$. We further interpret this formula as the Hilbert function of the special fiber ring $\mathcal{F}(I)$, compute the analytic spread, and prove that $\mathcal{F}(I)$ is a normal Cohen--Macaulay affine semigroup ring. Finally, we show that $I$ is matroidal and that every ordinary power $I^n$ is polymatroidal; consequently, $I^n$ has linear quotients and a $2n$-linear minimal free resolution for all $n\geq 1$.