On Ordered Domains of Integrity

In a recent paper,1 T. Szele proved that a division ring D is orderable if and only if the additive and multiplicative semigroup 5 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*=P\J(-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 au ■ • • , an in K* such that a is a product of the 2« elements fli, • • • , an, au • • ■ , an in some order. We denote by 5 the additive semigroup generated by the even elements of K*. The theorem we wish to prove is as follows.