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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. A ring R is called an O∗ring if each of its ring partial orders can be extended to a total order of R. Two of the problems in the list at the back of Fuch’s book [4] concern O∗rings. (A) Establish ring theoretical properties of O∗rings. (B) Does there exist a polynomial identity which forces each totally ordered ring that satisfies it to be an O∗ring? These problems were perhaps motivated by the well-known fact that each torsionfree abelian group is an O∗group. Recently, Kreinovich [7] has shown that (B) has a negative answer in the sense that if f(x1, · · · , xn) = 0 is such an identity, then each O∗ring that satisfies it must be trivial; that is, R = 0. In the process of showing this he noted that an O∗ring has two very restrictive properties: it is algebraic over Z and each nilpotent element has index at most two. To see this first recall that the partial order in a partially ordered ring R is determined by its positive cone R = {x ∈ R : x ≥ 0}; we will refer to such a positive cone as a partial order of R. Now if a is an element of an O∗ring R that is not algebraic over Z, then Z+[−a2] is a partial order of R which is not contained in any total order of R. Also, if a ∈ R is nilpotent of index n > 2 let b = −an−2 if n is even and let b = −an−1 if n is odd. Then Zb is a partial order of R that is not contained in any total order of R. Clearly, each subring of an O∗ring R is an O∗ring, and its divisible hull d(R) is also an O∗ring. For if P is a partial order of d(R) and T is a total order of R which contains P ∩R, then d(T ) = {x ∈ d(R) : ∃n > 0 with nx ∈ T } is a total order of d(R) which contains P . Consequently, in this paper we will deal exclusively with algebras over the rationals Q. All such O∗algebras are determined in the Theorem. If R is an O∗algebra, then there is a Q vector space N such that R is (isomorphic to) one of the following algebras. Received by the editors April 9, 1996. 1991 Mathematics Subject Classification. Primary 06F25. c ©1997 American Mathematical Society

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