Obstructing extensions of the functor Spec to noncommutative rings

TL;DR

This paper investigates the limitations of extending the prime spectrum functor from commutative to noncommutative rings, showing obstructions related to matrix rings and employing quantum mechanics principles.

Contribution

It proves that certain functors extending Spec to noncommutative rings must assign empty sets to matrix rings of size three or more, revealing fundamental obstructions.

Findings

Functors extending Spec assign empty sets to M_n(C) for n >= 3.

Uses Kochen-Specker Theorem to establish obstructions.

Shows similar obstructions for Gelfand spectrum extensions in C*-algebras.

Abstract

In this paper we study contravariant functors from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to the prime spectrum functor Spec. The main result reveals a common characteristic of these functors: every such functor assigns the empty set to M_n(C) for n >= 3. The proof relies, in part, on the Kochen-Specker Theorem of quantum mechanics. The analogous result for noncommutative extensions of the Gelfand spectrum functor for C*-algebras is also proved.