Sylvester rank functions for amenable normal extensions

We introduce a notion of amenable normal extension S unital ring R with finite approximation system F, encompassing the algebras over field Gromov and Elek, twisted crossed product by an group, tensor extension. It is shown that every Sylvester matrix rank function rk preserved has canonical to rk_F for S. In case extension, it also depends on continuously. When group action C*-algebra preserving tracial state, we show two natural functions algebraic constructed out state coincide.