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 $\text{EL}\_n(\mathbf{Z}[X\_1,\ldots,X\_m]) = \text{SL}\_n(\mathbf{Z}[X\_1,\ldots,X\_m])$?



Why do elementary matrices generate the special linear group over polynomial rings?
This is the "Suslin stability theorem". There is an [algorithmic proof](http://www.math.uiuc.edu/K-theory/0019/stability.pdf) in Park+ Woodburn, "An algorithmic proof of the Suslin stability theorem for polynomial rings."


