Let $D$ be a division ring and $R=D[t_1,\ldots,t_n]$ the polynomial ring in $n$ variables. Now let $GL_m(R),\,E_m(R)$ be the usual general linear group and its subgroup generated by the elementary matrices of the form $I_m+r\epsilon(i,j)$ ($r\in R$). Further, by $U(R)$ I mean the units of $R$. This may be confused with $GL_1(R)$ in a natural way. My question is this, when is $GL_m(R)=U(R)\cdot E_m(R)$, i.e. when can any $X\in GL_m(R)$ be written as a product of elementary matrices (up to a unit)?

I know that, by a result of Cohn, that $GL_2(R)\neq U(R)\cdot E_2(R)$, but what about the case when $m>2$? If $D$ was commutative then we could obviously apply Suslin's stability theorem, but as far as I know this is not true for the case of division rings.

andit is not too late to fix this. $\endgroup$11more comments