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 ? 



Writing a matrix as a sum of two invertible matrices
**The answer is negative.**


There is a nice theorem of **M. Henriksen** which says that **If $n\geq 2$ then every element of $M\_n(R)$ is a sum of three units** also he proves that there are non-unit matrices in $\bf{M\_2(\Bbb{Z}\_2[x\_1,x\_2])}$ that can not be written as a sum of two units. You can find a copy of the article [HERE](http://www.math.hmc.edu/~henriksen/publications/1974_Henriksen_Two_classes_of_rings_generated_by_their_units.pdf) 


