Algebraically closed commutative local rings

Abstract

A commutative ring R with identity is called a local ring if R has only one maximal ideal. This is equivalent to saying that the sum of two nonunits is a non-unit. Therefore the theory of all commutative local rings is axiomatizible by a finite set of A2-sentences. A commutative local ring with identity is said to be an algebraically closed local ring if every finite system of polynomial equations and inequations in one or more variables with coefficients in R which has a solution in some commutative local extension of R already has a solution in R. Much work connected with algebraically closed structures of classes of rings has been done, for example by Cherlin [2], Macintyre [4] and Lipschitz and Saracino [3]. We want to show similar results for commutative local rings with identity. Our main results are the following:Theorem. The theory of commutative local rings with identity has no model-companion.The finitely generic and infinitely generic local rings are algebraically closed local rings.Theorem. There is an A3 sentence which holds for all finitely generic local rings whose negation holds in every infinitely generic local ring.