Corollary

The dual of a finitely generated projective module is a finitely generated projective module.

A finitely generated projective left A-module can be identified with a direct factor M of a free A-module $A_s^n$ with a finite basis (no. 2, Corollary 1 to Proposition 4). Then (Proposition 11 and Corollary to Proposition 10) M* is isomorphic to a direct factor of $A_d^n$, whence the corollary.