The universal abelian regular ring

Olivier’s construction of a universal commutative regular ring is generalized to obtain for a noncommutative ring R the universal abelian regular R-ring R → Rˆ. This R-ring induces a homeomorphism from the spectrum of Rˆ to the constructible Cohn spectrum of R. An ́etale bundle is constructed over the Ziegler spectrum Zg∗(R), equipped with the Zariski topology, whose sheaf of sections is the sheafification of Prest’s presheaf of definable scalars. When the bundle is restricted to Zg∗(Rˆ), which is homeomorphic to the spectrum of Rˆ, one obtains an ́etale bundle whose R-ring of global sections is isomorphic to Rˆ.