Algebraically closed commutative rings

Abstract

A commutative ring A is said to be algebraically closed if every finite system of polynomial equations and inequations in one or more variables with coefficients in A which has a solution in some (commutative) extension of A already has a solution in A. Abraham Robinson's study of model-theoretic forcing has provided powerful new tools for the study of algebraically closed structures in general, and will be applied here to the study of algebraically closed commutative rings. Familiarity with the model-theoretic notions connected with the study of algebraically closed structures is assumed; for background consult [1], [2], and [3].Our main results are the following:1. The theory of commutative rings with identity has no model companion in the sense of Robinson.2. The Hilbert Nullstellensatz, suitably formulated for the class of algebraically closed commutative rings, holds for finitely generated polynomial ideals but fails for certain infinitely generated polynomial ideals.3. If A is algebraically closed, then A/rad A need not be algebraically closed as a semiprime ring: If A is finitely generic then A/rad A is algebraically closed as a semiprime ring, but if A is infinitely generic then A/rad A is not algebraically closed as a semiprime ring.