TITLE: Writing a matrix as a sum of two invertible matrices
QUESTION [13 upvotes]: 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 ?

REPLY [19 votes]: 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