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The prime ideal graph $\\Gamma_P(R)$ 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 $\\Gamma_P(R)\\cong K_{m-1}\\vee \\overline{K}_{m(q-1)}$. 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We also prove a realization theorem showing that every complete split graph of this form arises from a prime ideal of a finite commutative ri"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that Γ_P(R) ≅ K_{|P|-1} ∨ K-bar_{|R|-|P|}, and a monomial x^α y^β belongs to the minimal generators of I(Γ_P(R))^n if and only if |α| + |β| = 2n, |β| ≤ n, and 0 ≤ α_i ≤ n for all i.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"R is a finite commutative ring with identity and P is a proper prime ideal; this finiteness and the prime property are used to obtain the explicit isomorphism and the counting formulas.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Prime ideal graphs are complete split graphs whose edge ideal powers have explicit minimal monomial generators satisfying exponent conditions, are polymatroidal with linear resolutions, and have computable analytic spread.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Prime ideal graphs of finite rings are complete split graphs whose edge ideal powers have explicit generators and are polymatroidal.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b23fee55ffd435a6cdfc28ef95459968db64efed4caef555970eede29f72c120"},"source":{"id":"2604.19408","kind":"arxiv","version":3},"verdict":{"id":"d1313ddd-4dff-418a-9836-333f16bdeb8f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T21:37:31.042953Z","strongest_claim":"We prove that Γ_P(R) ≅ K_{|P|-1} ∨ K-bar_{|R|-|P|}, and a monomial x^α y^β belongs to the minimal generators of I(Γ_P(R))^n if and only if |α| + |β| = 2n, |β| ≤ n, and 0 ≤ α_i ≤ n for all i.","one_line_summary":"Prime ideal graphs are complete split graphs whose edge ideal powers have explicit minimal monomial generators satisfying exponent conditions, are polymatroidal with linear resolutions, and have computable analytic spread.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"R is a finite commutative ring with identity and P is a proper prime ideal; 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