Real algebraic geometry for matrices over commutative rings

We define and study preorderings and orderings on rings of the form Mn(R) where R is a commutative unital ring. We extend the Artin–Lang theorem and Krivine–Stengle Stellensätze (both abstract and geometric) from R to Mn(R). This problem has been open since the seventies when Hilbertʼs 17th problem was extended from usual to matrix polynomials. While the orderings of Mn(R) are in one-to-one correspondence with the orderings of R, this is not true for preorderings. Therefore, our theory is not Morita equivalent to the classical real algebraic geometry.