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We show that under some conditions a Gorenstein ring 
$R$ satisfies the Generalized Auslander–Reiten conjecture if and only if 
$R\left[ x \right]$ does. When $R$ is a local ring we prove the same result for some localizations of $R\left[ x \right]$.
The class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms 
τtl. 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 
oτl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τtl is companionable, with model-companion oτl.
