[Joint work with F. Miraglia, Univ. of Sao Paulo, Brazil]
I'll outline a proof of Milnor's Witt ring conjecture and of Marshall's signature conjecture for the (reduced) theory of quadratic forms on rings of the type mentioned in the title. The proof is carried out in the context of (reduced) special groups --an axiomatic theory of quadratic forms--, and is an application of k-theoretic techniques in this context (developed in a forthcoming paper in Journal of Pure and Applied Algebra), which I will explain in the talk. The proof also resorts to our solution of Lam's conjecture for preordered fields.