Cut structures in zero-divisor graphs of commutative rings

Axtell, Mike; Baeth, Nicholas R.; Stickles, Joe

doi:10.1216/jca-2016-8-2-143

Cited by 6 publications

(1 citation statement)

References 16 publications

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“…The graph Γ E (R) was further studied by D.F. Anderson and J. LaGrange [2] (also see [3]), S. Spiroff and C. Wickham [12], M. Axtell, N. Baeth, and J. Stickles [6]. For a detailed and nice survey of the zerodivisor graphs with a special emphasis to Γ E (R), see J. Coykendall et al [8].…”

Section: Introductionmentioning

confidence: 99%

On the problem of the finiteness of the compressed zero divisor graphs of Artinian rings

Kadu¹

2020

Preprint

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Let R be an Artinian ring and let Γ E (R) be the compressed zero-divisor graph associated to R. The question of when the clique number ω(Γ E (R)) < ∞ was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff, see [8, Section 5]. They proved that if ℓ(R) ≤ 4 thenthey gave an example of a local ring R where ω(Γ E (R)) = ∞ is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when ℓ(R) = 5 was stated as an open problem. We show that if ℓWe first reduce the problem to the case of a local ring (R, m, k). We then enumerate all possible Hilbert functions of R and show that the k-vector space m/m 2 admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space m/m 2 with the structure of zero-divisors in R. For instance, in the case when m 2 is principal and m 3 = 0, we show that R is Gorenstein if and only if the symmetric bilinear form on m/m 2 is non-degenerate. Moreover, in the case when ℓ(R) = 4, our techniques also yield a simpler and shorter proof of the finiteness of ω(Γ E (R)) avoiding, for instance, Cohen structure theorem.

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Section: Introductionmentioning

confidence: 99%