Under what condition can any $X\in GL_2(R)$ be reduced to a triangular matrix?

Question

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.

Answer 1

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?