Edge Ideals of Prime Ideal Graphs: Ordinary Powers, Polymatroidality, and Analytic Spread
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| Format: | Preprint |
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2026
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| _version_ | 1866909038895169536 |
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| author | Rasheed, Tabinda Yao, Wang |
| author_facet | Rasheed, Tabinda Yao, Wang |
| contents | 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 of $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy\in P$. We prove 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. Using this description, we determine the minimal vertex covers and obtain an irredundant primary decomposition of the edge ideal $I(Γ_P(R))$. For every $n\geq 1$, we characterize the minimal monomial generators of the ordinary power $I(Γ_P(R))^n$: a monomial $x^αy^β$ belongs to $G(I(Γ_P(R))^n)$ if and only if $|α|+|β|=2n, \ |β|\leq n$, and $0\leq α_i\leq n$ for all $i$. Consequently, we derive a closed formula for $μ(I(Γ_P(R))^n)$. We also prove that every ordinary power is polymatroidal and hence has linear quotients and a $2n-$linear resolution. Finally, we interpret $μ(I(Γ_P(R))^n)$ as the Hilbert function of the special fiber ring and compute the analytic spread of $I(Γ_P(R))$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_19408 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Edge Ideals of Prime Ideal Graphs: Ordinary Powers, Polymatroidality, and Analytic Spread Rasheed, Tabinda Yao, Wang Commutative Algebra Combinatorics 2020: 13F55, 05C25, 05E40, 13A15 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 of $R\setminus\{0\}$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy\in P$. We prove 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. Using this description, we determine the minimal vertex covers and obtain an irredundant primary decomposition of the edge ideal $I(Γ_P(R))$. For every $n\geq 1$, we characterize the minimal monomial generators of the ordinary power $I(Γ_P(R))^n$: a monomial $x^αy^β$ belongs to $G(I(Γ_P(R))^n)$ if and only if $|α|+|β|=2n, \ |β|\leq n$, and $0\leq α_i\leq n$ for all $i$. Consequently, we derive a closed formula for $μ(I(Γ_P(R))^n)$. We also prove that every ordinary power is polymatroidal and hence has linear quotients and a $2n-$linear resolution. Finally, we interpret $μ(I(Γ_P(R))^n)$ as the Hilbert function of the special fiber ring and compute the analytic spread of $I(Γ_P(R))$. |
| title | Edge Ideals of Prime Ideal Graphs: Ordinary Powers, Polymatroidality, and Analytic Spread |
| topic | Commutative Algebra Combinatorics 2020: 13F55, 05C25, 05E40, 13A15 |
| url | https://arxiv.org/abs/2604.19408 |