A Kadison–Dubois representation for associative rings

In this paper we give a general theorem that describes necessary and sufficient conditions for a module to satisfy the so-called Kadison–Dubois property. This is used to generalize Jacobiʹs version of the Kadison–Dubois representation to associative rings. We apply this representation to obtain a noncommutative algebraic and geometric version of PutinarʹsPositivstellensatz. We finish the paper by answering questions given by Marshall and Jacobi.