Orderings of higher level on noetherian rings

The noncommutative algebraic geometry has found fruitful applications in quantum geometry. Similar applications are expected to be found for its younger sister the noncommutative real algebraic geometry

One of the basic results in real algebraic geometry is the Positivestellensatz. The original results of Dubois and Risler (see section 3.3 of 13]) have been extended in many directions. We refer to 14], 1], 2], 3] for commutative rings and 9], 4] for associative rings. The aim of this paper is to prove the higher level Posit ivstellensatz for noncommutative Noetherian rings. Our proof depends on the intersection theorem for orderings of higher level on skew fields (11], Theorem 3.13). The general case of orderings of higher level on associative rings remains open.