Abstract

Let R be a commutative Artinian ring and let ${\Gamma }_E(R)$ be the compressed zero-divisor graph associated to R. The question of when the clique number $\omega ({\Gamma }_E(R))<\infty$ was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff. They proved that if $\mathcal{\ell }(R)\le 4$ (where $\mathcal{\ell }(R)$ is the largest length of any of its chains of ideals), then $\omega ({\Gamma }_E(R))<\infty .$ When $\mathcal{\ell }(R)=6,$ they gave an example of a local ring R where $\omega ({\Gamma }_E(R))=\infty$ is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when $\mathcal{\ell }(R)=5$ was stated as an open question. We show that if $\mathcal{\ell }(R)=5$ then $\omega ({\Gamma }_E(R))<\infty .$ We first reduce the problem to the case of a local ring $(R,\mathfrak{m},k).$ We then enumerate all possible Hilbert functions of R and show that the k-vector space $\mathfrak{m}/{\mathfrak{m}}^2$ admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space $\mathfrak{m}/{\mathfrak{m}}^2$ with the structure of zero-divisors in R. For instance, in the case when ${\mathfrak{m}}^2$ is principal and ${\mathfrak{m}}^3=0,$ we show that R is Gorenstein if and only if the symmetric bilinear form on $\mathfrak{m}/{\mathfrak{m}}^2$ is non-degenerate. Moreover, in the case when $\mathcal{\ell }(R)=4,$ our techniques also yield a simpler and shorter proof of the finiteness of $\omega ({\Gamma }_E(R))$ avoiding, for instance, the Cohen structure theorem.

摘要

令R为可交换的 Artinian 环，并令${\Gamma }_{乙}(\mathrm{电阻})$是与R关联的压缩零除数图。团号什么时候出的问题$\omega ({\Gamma }_{乙}(\mathrm{电阻}))<\infty$由 J. Coykendall、S. Sather-Wagstaff、L. Sheppardson 和 S. Spiroff 抚养长大。他们证明，如果$\mathcal{\ell }(\mathrm{电阻})\le 4$ （在哪里 $\mathcal{\ell }(\mathrm{电阻})$ 是其任何理想链的最大长度），然后 $\omega ({\Gamma }_{乙}(\mathrm{电阻}))<\infty .$ 什么时候 $\mathcal{\ell }(\mathrm{电阻})=6,$他们给出了一个本地环R的例子，其中$\omega ({\Gamma }_{乙}(\mathrm{电阻}))=\infty$通过其二元化模块使用 Artinian 局部环的平凡扩展是可能的。什么时候发生的问题$\mathcal{\ell }(\mathrm{电阻})=5$被表述为一个开放性问题。我们证明如果$\mathcal{\ell }(\mathrm{电阻})=5$ 然后 $\omega ({\Gamma }_{乙}(\mathrm{电阻}))<\infty .$ 我们首先将问题归结为局部环的情况 $(\mathrm{电阻},\mathfrak{米},克).$然后我们枚举R 的所有可能的希尔伯特函数并证明k向量空间$\mathfrak{米}/{\mathfrak{米}}^2$在 Hilbert 函数的某些情况下允许对称双线性形式。这允许我们关联双线性空间中的正交性$\mathfrak{米}/{\mathfrak{米}}^2$与R 中的零除数结构。例如，在这种情况下${\mathfrak{米}}^2$ 是主要的和 ${\mathfrak{米}}^3=0,$我们证明R是 Gorenstein 当且仅当对称双线性形式$\mathfrak{米}/{\mathfrak{米}}^2$是非退化的。此外，在这种情况下$\mathcal{\ell }(\mathrm{电阻})=4,$ 我们的技术还产生了一个更简单和更短的证明的有限性 $\omega ({\Gamma }_{乙}(\mathrm{电阻}))$ 例如，避免 Cohen 结构定理。