A characterization of rings in which each partial order is contained in a total order

Abstract:

Rings in which each partial order can be extended to a total order are called $O^\ast$- rings by Fuchs. We characterize $O^\ast$- rings as subrings of algebras over the rationals that arise by freely adjoining an identity or one-sided identity to a rational vector space $N$ or by taking the direct sum of $N$ with an $O^\ast$- field. Each real quadratic extension of the rationals is an $O^\ast$- field.