A quantitative general Nullstellensatz for Jacobson rings

Abstract

The general Nullstellensatz states that if $A$ is a Jacobson ring, $A[X]$ is Jacobson. We introduce the notion of an $\alpha$-Jacobson ring for an ordinal $\alpha$ and prove a quantitative version of the general Nullstellensatz: if $A$ is an $\alpha$-Jacobson ring, $A[X]$ is $(\alpha+1)$-Jacobson. The quantitative general Nullstellensatz implies that $K[X_1,\ldots,X_n]$ is not only Jacobson but also $(1+n)$-Jacobson for any field $K$. It also implies that $\mathbb{Z}[X_1,\ldots,X_n]$ is $(2+n)$-Jacobson.