A constructive proof of the general Nullstellensatz for Jacobson rings

TL;DR

This paper provides a constructive proof that univariate polynomial rings over Jacobson rings are Jacobson, resolving an open problem and extending the Nullstellensatz to finitely Jacobson rings.

Contribution

It offers a constructive proof of the general Nullstellensatz for Jacobson rings and its variants, addressing an open problem in constructive algebra.

Findings

Univariate polynomial rings over Jacobson rings are Jacobson.

Finitely generated algebras over zero-dimensional rings are Jacobson.

Extended Nullstellensatz for finitely Jacobson rings.

Abstract

We give a constructive proof of the general Nullstellensatz: a univariate polynomial ring over a commutative Jacobson ring is Jacobson. This theorem implies that every finitely generated algebra over a zero-dimensional ring or the ring of integers is Jacobson, which has been an open problem in constructive algebra. We also prove a variant of the general Nullstellensatz for finitely Jacobson rings.