Let $R$ be a commutative ring. Let $E_n (R)$ be the subgroup of $GL_n(R)$ generated by all matrices of the form $I + \lambda$ where $\lambda $ is a matrix with precisely one non zero entry and this entry does not occur on the diagonal. Suppose that $R$ is a euclidean ring. Show that $SL_n(R) = E_n(R)$ (where $SL_n (R) $ are matrices with determinant $1$).
I know how to prove this for $R$ is a field, by adding and subtracting some multiple of rows and columns and proceed by induction. But how to do this for a euclidean ring? Any help is appreciated.