On ordered domains of integrity

Abstract

In a recent paper,' T. Szele proved that a division ring D is orderable if and only if the additive and multiplicative semigroup S generated by the nonzero squares of elements of D does not contain the zero element of D. The present paper extends this result to a domain of integrity K. Let us denote by K* the set of nonzero elements of K. The domain of integrity K is said to be orderable if and only if there exists an additive and multiplicative semigroup P (the positive elements) contained in K* such that K*=PUJ(-P). If K does not have a unit element, then there exists a unique minimal domain of integrity K having a unit element and containing K.2 It is not too difficult to show that K is orderable if and only if K is orderable. For this reason, we assume henceforth that K has a unit element. An element a of K* is called even if there exist elements a,, ,an in K* such that a is a product of the 2n elements a,, an, a,, * *, an in some order. We denote by S the additive semigroup generated by the even elements of K*. The theorem we wish to prove is as follows.