Suppose $R$ is a (possibly noncommutative) ring. I was thinking of $R=S[x_1,\ldots,x_n]$ or $R=S[x_1,x_1^{-1},\ldots,x_n,x_n^{-1}]$ for $S$ some (possibly noncommutative) ring. Now, let $GL_2(R)$ be the group of invertible matrices over $R$, and $E(R)$ the subgroup of $GL_2(R)$ generated by matrices of the form $I_2+r\epsilon(i,j)$ ($r\in R$).

Are there any conditions upon $R$ (or $S$) that would allow us to 'reduce' any matrix $X\in GL_2(R)$ to $X=TE$, where $T$ is triangular, and $E\in E_2(R)$? In other words, can we multiply any $X\in GL_2(R)$ by elementary matrices until we get a triangular matrix?

This is clearly related to the work of Cohn and $GE$ rings; that is, rings such that $X=UE$ for $U\in U(R)$ and $E\in E_2(R)$. However, the property I am after seems much more attainable that this 'GE' property.