The Real Spectrum of a Noncommutative Ring and the Artin-Lang Homomorphism Theorem

Let R be a noncommutative ring. Two epimorphisms

$$alpha_{i}:Rto (D_{i},leqslant_{i}),quad i = 1,2 $$

from R to totally ordered division rings are called equivalent if there exists an order-preserving isomorphism ? : (D ₁, ? ₁) → (D ₂, ? ₂) satisfying ? ° α ₁ = α ₂. In this paper we study the real epi-spectrum of R, defined to be the set of all equivalence classes (with respect to this relation) of epimorphisms from R to ordered division rings. We show that it is a spectral space when endowed with a natural topology and prove a variant of the Artin-Lang homomorphism theorem for finitely generated tensor algebras over real closed division rings.