Projective modules over Rees-like algebras and its monoid extensions

Let A be a Rees-like algebra of dimension d and N a commutative partially cancellative torsion-free seminormal monoid. We prove the following results. enumerate Let P be a finitely generated projective A-module of d. Then (i) P has a unimodular element; (ii) The action of (A P) on (A P) is transitive. Let P be a finitely generated projective AN-module of ~r. Then (i) P has a unimodular element for r\3, d\; (ii) The action of (AN P) on (AN P) is transitive for r\2, d\. enumerate These improve the classical results of Serre Se58 and Bass Ba64.