  The class of all Artinian local rings of length at most l is A_2-elementary,
axiomatised by a finite set of axioms Art_l. We show that its existentially
closed models are Gorenstein, of length exactly l and their residue fields are
algebraically closed, and, conversely, every existentially closed model is of
this form. The theory Gor_l of all Artinian local Gorenstein rings of length l
with algebraically closed residue field is model complete and the theory Art_l
is companionable, with model-companion Gor_l.
