Let $R$ be a ring. An *elementary matrix* over $R$ is a matrix with $1$s along the diagonal and at most one other nonzero entry. Let $\text{EL}_n(R)$ denote the subgroup of $\text{GL}_n(R)$ generated by the elementary matrices.

I understand that $\text{EL}_n(R) = \text{SL}_n(R)$ provided that $R$ is a Euclidean domain.

Why does $\text{EL}_n(\mathbf{Z}[X_1,\ldots,X_m]) = \text{SL}_n(\mathbf{Z}[X_1,\ldots,X_m])$?