On the problem of zero-divisors in Artinian rings

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.