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 * -rings by Fuchs. We characterize O * -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 *field. Each real quadratic extension of the rationals is an O * -field.