On the problem of zero-divisors in Artinian rings

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.

Keywords:

• Artinian rings
• zero divisor graphs

2020 Mathematics Subject Classification:

• Primary: 13A70, 05E40
• Secondary: 05C69, 13E39

Acknowledgments

We thank the referee for many pertinent comments, in particular, for pointing out a simpler proof of the Lemma 2.2. Example 2.3 is also due to referee. It is a pleasure to thank Vinayak Joshi for many useful discussions on the subject of this paper.

Additional information

Funding

The author thanks DST-SERB for financial assistance under the project ECR/2017/000790, Department of Science and Technology, Science and Engineering Research Board, Government of India.