Powers of Edge Ideals of Prime Ideal Graphs: Generators, Primary Decomposition, and Linear Powers

Abstract: Let $R$ be a finite commutative ring with multiplicative identity. Let $P$ be a proper prime ideal of $R$. The prime ideal graph $ΓP(R)$ has a vertex set of $R\setminus{0}$, and two different vertices $x$ and $y$ are adjacent to each other if and only if $xy\in P$. We show that $Γ_P(R)\cong K{|P|-1}\vee \overline{K}_{\,|R|-|P|}$, so prime ideal graphs form a ring-induced family of complete split graphs determined by the two parameters $a=|P|-1$ and $b=|R|-|P|$. Using this decomposition, we determine the minimal vertex covers of $Γ_P(R)$, obtain an explicit irredundant primary decomposition of the edge ideal $I(Γ_P(R))$, and characterize the minimal monomial generators of every power $I(Γ_P(R))^n$. More precisely, $x^αy^β\in G(I(Γ_P(R))^n)$ if and only if $|α|+|β|=2n, \ |β|\leq n$, and $0\leq α_i\leq n$ for all $i$. As a consequence, we derive a closed formula for $μ(I(Γ_P(R))^n)$ and prove that every power $I(Γ_P(R))^n$ is polymatroidal. Hence, each power has linear quotients and a $2n-$linear minimal free resolution; equivalently, $I(Γ_P(R))$ has linear powers. Examples over $\mathbb{Z}_6$ and $\mathbb{Z}_8$ are included.

• The paper presents an explicit parametrization of the minimal monomial generators of the n-th power of the edge ideal, linking ring properties with split graph combinatorics.
• It derives an irredundant primary decomposition that distinguishes the roles of clique and independent set components in the prime ideal graphs.
• The results prove all powers of these edge ideals are polymatroidal, ensuring linear resolutions with regularity equal to twice the power index.

Edge Ideals of Prime Ideal Graphs: Structure, Generators, and Homological Properties

Introduction

This paper investigates the algebraic and combinatorial structure of edge ideals associated with prime ideal graphs $\Gamma_P(R)$, where $P$ is a proper prime ideal of a finite commutative ring $R$. The study situates itself at the intersection of combinatorial commutative algebra and algebraic graph theory, extending classical results on edge ideals and their powers to a ring-theoretic family of split graphs with explicit parameterization by the sizes of $P$ and $R$.

Prime Ideal Graphs as Complete Split Graphs

The prime ideal graph $\Gamma_P(R)$ is constructed by placing an edge between each pair of nonzero ring elements whose product lies in $P$. The analysis establishes an isomorphism

$$\Gamma_P(R) \cong K_{|P|-1} \vee \overline{K}_{|R|-|P|}$$

where $K_{|P|-1}$ is a clique corresponding to the nonzero elements of $P$, and $P$0 is an independent set formed by $P$1. The graph join imparts a split graph structure with clique and independent set sizes governed directly by the underlying ring and ideal.

This structural perspective enables a direct translation of the combinatorial properties of $P$2 into algebraic properties of its edge ideal. The maximal cliques, minimal vertex covers, and chromatic number are parameterized canonically in terms of $P$3 and $P$4.

Primary Decomposition and Minimal Generators

The paper derives an explicit irredundant primary decomposition for the edge ideal $P$5 (with $P$6 and $P$7). The associated primes correspond to the natural set partitions induced by $P$8 and $P$9. Each associated prime is either supported on the clique or on the union of nearly all clique variables with the entire independent set, reflecting the split graph geometry.

Further, the paper gives a complete characterization of the minimal monomial generators of the $R$0th power $R$1, showing that a monomial $R$2 is a generator if and only if $R$3, $R$4, and no exponent $R$5. This explicit parametrization leads to a closed formula for the number of minimal generators: $R$6 where $R$7 and $R$8 are as defined. This formula is novel in its ring-parameterized, non-asymptotic nature and establishes a strong connection between the combinatorics of split graphs and the algebraic invariants of their edge ideals.

Polymatroidal Powers and Homological Consequences

A central result is that for all $R$9, the power $P$0 is polymatroidal. The proof utilizes the explicit description of the generators and exploits the exchange property between the $P$1 and $P$2 exponent vectors. As a consequence, all powers possess linear quotients and thus admit $P$3-linear minimal free resolutions. Equivalently, $P$4 has linear powers, i.e., $P$5 for all $P$6.

The identification of a direct polymatroidal structure for all powers is significant, as it circumvents reliance on more abstract or asymptotic arguments. The invariants—height, unmixedness, and Krull dimension—are calculated explicitly: the ideal is not unmixed except in the trivial complete graph case.

Example Computations and Explicit Families

Illustrative examples are provided for $P$7 and $P$8 with explicit computation of generators and enumeration of $P$9 for small $R$0. For instance, in the case $R$1, $R$2, the associated split graph has $R$3, $R$4, and the second power $R$5 has $R$6 minimal generators and a $R$7-linear resolution. The examples further demonstrate the general formula for the family $R$8, elucidating asymptotic behavior as $R$9 and $\Gamma_P(R)$0 grow.

Theoretical and Practical Implications

The results clarify the interplay between commutative algebra and the combinatorics of ring-based split graphs. By parameterizing split graph edge ideals in terms of ring-theoretic data, the results enable explicit formulae for homological invariants that, for arbitrary split graphs, are typically inaccessible without further structural constraints. The structural results provide new computational tools for understanding resolutions, associated primes, and generator growth in large classes of edge ideals relevant both in algebraic combinatorics and in coding-theoretic applications where finite rings play a role.

Avenues for future work include extending the framework to symbolic powers, determining the full set-theoretic associated primes for higher powers, and computing graded Betti tables. The polymatroidal structure and explicit generator parameterization may facilitate advances in the study of test ideals, regularity jumps, and depth stability for edge ideals arising from ring-induced graphs.

Conclusion

The paper provides a comprehensive analysis of the powers of edge ideals of prime ideal graphs, characterized as complete split graphs arising from ring-theoretic constructions. The explicit descriptions of generators, primary decompositions, and the polymatroidal property for all powers establish a rigorous foundation for the study of such ideals. The results integrate structural, combinatorial, and homological perspectives and open further questions concerning more refined invariants and symbolic powers in this ring-induced graph setting.