Abstract
Let R be a commutative Artinian ring and let be the compressed zero-divisor graph associated to R. The question of when the clique number
was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff. They proved that if
(where
is the largest length of any of its chains of ideals), then
When
they gave an example of a local ring R where
is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when
was stated as an open question. We show that if
then
We first reduce the problem to the case of a local ring
We then enumerate all possible Hilbert functions of R and show that the k-vector space
admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space
with the structure of zero-divisors in R. For instance, in the case when
is principal and
we show that R is Gorenstein if and only if the symmetric bilinear form on
is non-degenerate. Moreover, in the case when
our techniques also yield a simpler and shorter proof of the finiteness of
avoiding, for instance, the Cohen structure theorem.
Keywords:
2020 Mathematics Subject Classification:
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.