The Hermite ring conjecture and special linear groups for valuation rings

In this paper we prove that the Hermite ring conjecture holds for valuation rings $V$, and the special liner group $S{L}_n(V[x])$ coincides with the group generated by elementary matrices $E_n(V[x])$ for $n\ge 3$. For any arithmetical ring $R$ and $n\ge 3$, we show $S{L}_n(R[x])=S{L}_n(R)\cdot E_n(R[x])$.