Abstract

The class of all Artinian local rings of length at most l is $\forall_2$-elementary, axiomatised by a finite set of axioms $\mathscr{A}\mathfrak{rt}_l$. 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 $\mathscr{G}\mathfrak{ot}_l$ of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory $\mathscr{A}\mathfrak{rt}_l$ is companionable, with model-companion $\mathscr{G}\mathfrak{ot}_l$.