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 $A[N]$-module of ~r. Then $(i)$ P has a unimodular element for r≥max\3,d\; $(ii)$ The action of (A[N]⊕ P) on (A[N]⊕ P) is transitive for r≥max\2,d\. {enumerate} These improve the classical results of Serre {Se58} and Bass {Ba64}.