Existentially closed models of the theory of artinian local rings

Journal of Symbolic Logic 64 (2):825-845 (1999)
Abstract
The class of all Artinian local rings of length at most l is ∀ 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 Got 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 Got l
Keywords Model Theory   Existentially Closed Models   Artinian Local Rings   Gorenstein Rings
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