The Central Nullstellensatz over Centrally Algebraically Closed Division Rings

TL;DR

This work develops a noncommutative analogue of Hilbert’s Nullstellensatz for division rings by introducing centrally algebraically closed (centrally AC) division rings and proving that the central Nullstellensatz holds if and only if $D$ is centrally AC. The authors connect algebraicity over centralizers to a robust framework, using the Amitsur–Small theorem to link finite-dimensional module behavior to a central-geometry-like Nullstellensatz. They establish a Hilbert-style Nullstellensatz for centrally AC division rings and show that a division ring is a Nullstellensatz ring exactly when it is centrally AC, with a comprehensive discussion of centralizers and rational identities. The results yield an embedding result, show that every division ring can sit inside a centrally AC division ring, and open several questions about existential closure and broader closures in noncommutative settings, anchoring future research in noncommutative algebraic geometry. The framework highlights how centralizers govern algebraicity and Nullstellensatz phenomena in noncommutative rings, offering a path to generalize classical algebraic geometry to division rings.

Abstract

We introduce the concept of centrally algebraically closed division rings and show that a division ring satisfies the central Nullstellensatz if and only if it is centrally algebraically closed. We also show that every division ring can be embedded in a centrally algebraically closed division ring.

Theorems & Definitions (34)

• Theorem 1.1: Amitsur--Small Theorem
• Theorem 1.2: Alon--Paran Theorem
• Proposition 2.1
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Remark 3.1
• Proposition 3.3
• proof