  We investigate the standard graded $k$-algebras over a field $k$ of
characteristic zero for which general linear forms are exact zero divisors. We
formulate a conjecture regarding the Hilbert function of such rings. We prove
our conjecture in the case when the ring is a quotient of a polynomial ring by
a monomial idea, and also in the case when the ideal is generated in degree 2
and all but one of the generators are monomials.
