Writing a matrix as a sum of two invertible matrices Let $n\geq 2$. Is it true that any $n\times n$ matrix with entries from a given ring (with identity) can be written as a sum of two invertible matrices with entries from the same ring ? 

This may help. Any $2\times 2$ matrix is a sum of four units:
$$ \begin{pmatrix} x & y \\ z & t \end{pmatrix} = \begin{pmatrix} x & 1 \\ -1 & 0 \end{pmatrix} +  \begin{pmatrix} 0 & -1 \\ 1 & t \end{pmatrix} +  \begin{pmatrix} 1 & 0 \\ z & -1 \end{pmatrix} + \begin{pmatrix} -1 & y \\ 0 & 1 \end{pmatrix}$$
