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.



Under what condition can any $X\in GL_2(R)$ be reduced to a triangular matrix?
This an extended comment rather than an answer.


First of all, when working in the general linear group over a non-commutative ring, one should be very careful in translating the usual notions from the commutative setting. For example, consider a ring $R$ and two elements $x$ and $y$ such that $xy=1$ but $yx\neq1$. Then the matrix
$$ \begin{pmatrix} y & 1 \\ 0 & -x \end{pmatrix} $$
is invertible, and its inverse is
$$ \begin{pmatrix} x & 1 \\ 1-yx & -y \end{pmatrix}, $$
which is not upper triangular. Moreover, the diagonal entries of an invertible upper triangular matrix need not be invertible!


So one has to define the upper triangular subgroup $B$ as the subgroup of all upper triangular matrices having upper triangular inverses. Now this subgroup is subject to the *Levi decomposition*, which asserts that $B=D\ltimes U$. Here $D$ is the subgroup of diagonal matrices and $U$ is the subgroup of upper unit-triangular matrices (triangular with $1$ on the diagonal).


This said, you question turns out to be equivalent to your previous question [When is $GLm\_(R)$ generated by elementary and diagonal matrices?](https://mathoverflow.net/questions/258124/when-is-gl-mr-generated-by-elementary-and-diagonal-matrices)


