If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring

Abstract

In his classical 1963 book on partially ordered algebraic systems, L. Fuchs formulated the following problem (No. 29). It is known that if an abelian groupG (i.e., a group that satisfies an identityxy=yx) can be linearly ordered, then every partial order onG can be extended to a linear order. Fuchs asked whether there exists a similar polynomial identityP=0 for (associative) rings. In other words, does there exist a polynomialP with a following property: if a ringR satisfies the identityP=0, andR can be linearly ordered, then every partial order onR can be extended to a linear order?

We prove that no such non-trivial polynomial identity is possible. Namely, we prove that every ringR that satisfies such an identity is a zero-ring (i.e.,xy=0 for allx, y εR).