$K_1$-Stability of symplectic modules over monoid algebras

Let $R$ be a regular ring of dimension $d$ and $L$ be a $c$-divisible monoid. If $K_{1}Sp(R)$ is trivial and $k\ge d+2,$ then we prove that the symplectic group ${Sp}_{2k}(R[L])$ is generated by elementary symplectic matrices over $R[L]$. When $d\le 1$ or $R$ is a geometrically regular ring containing a field, then improved bounds have been established. We also discuss the linear case, extending the work of Gubeladze.