The intersection theorem for orderings of higher level in rings (Q1197367)

The main result of this paper is an intersection theorem for orderings of commutative rings of arbitrary level. Let \(A\) be a commutative ring with 1 and \(T\subset A\) a preordering of level \(n\). In Theorem 6 the author proves the following equivalency for \(a\in A\): 1) \(a\in P^ +\) for all orderings \(P\supset T\), where \(P^ +=P\setminus(P\cap-P)\), 2) \(at=1+t'\) for some \(t,t'\in T\). To obtain this result he first studies \(M\)-convex ideals of \(A\), where \(M\) is a \(T\)-module. As an application he proves a Positivstellensatz of higher level for a certain class of formally real fields.