Existence of Exact Zero Divisors and Totally Reflexive Modules in Artinian Rings

Abstract

In this dissertation, we consider commutative local Artinian rings (A, m, k), which are rings that satisfy the descending chain condition on ideals. First, we investigate the existence of exact zero divisors in Artinian Gorenstein rings. We say that a pair of elements (a, b) in m is an exact pair of zero divisors in A if ann(a) = (b) and ann(b) = (a). It is known that a generic Artinian Gorenstein ring of socle degree 3 contains at least one pair of exact zero divisors. We are interested in the existence of exact pairs of zero divisors in the case of socle degree bigger than 3. We present the conditions when an Artinian Gorenstein ring of socle degree bigger than 3 contains linear pairs of exact zero divisors.

We also investigate the existence of totally reflexive modules in the absence of exact pairs of zero divisors. Since the existence of totally reflexive modules is guaranteed in Artinian Gorenstein rings, we consider Artinian non-Gorenstein rings. We use Macaulay's Inverse System to construct these rings. As a result, we obtain a class of rings having non-free totally reflexive modules in the absence of exact pairs of zero divisors. We also discuss the Weak Lefschetz property and the connection between the Weak Lefschetz Property and exact pairs of zero divisors. We use Macaulay's Inverse System to construct rings in both investigations.