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. \begin{enumerate} \item Let $P$ be a finitely generated projective $A$-module of $\text{rank}\ge d$. Then $(i)$ $P$ has a unimodular element; $(ii)$ The action of $\text{EL}(A\oplus P)$ on $\text{Um}(A\oplus P)$ is transitive. \item Let $P$ be a finitely generated projective $A[N]$-module of $\text{rank}r$. Then $(i)$ $P$ has a unimodular element for $r\ge \max \{3,d\}$; $(ii)$ The action of $\text{EL}(A[N]\oplus P)$ on $\text{Um}(A[N]\oplus P)$ is transitive for $r\ge \max \{2,d\}$. \end{enumerate} These improve the classical results of Serre \cite{Se58} and Bass \cite{Ba64}.