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