  Let R be a local ring with maximal ideal m admitting a non-zero element
a\in\fm for which the ideal (0:a) is isomorphic to R/aR.
  We study minimal free resolutions of finitely generated R-modules M, with
particular attention to the case when m^4=0. Let e denote the minimal number of
generators of m. If R is Gorenstein with m^4=0 and e\ge 3, we show that \Poi
MRt is rational with denominator \HH R{-t} =1-et+et^2-t^3, for each finitely
generated R-module M. In particular, this conclusion applies to generic
Gorenstein algebras of socle degree 3.
