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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 ?

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    $\begingroup$ You might want to consider the answer in the $n=1$ case as an example of how you could handle the general case. I've voted to close. $\endgroup$
    – Ryan Budney
    Commented Sep 5, 2013 at 21:23
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    $\begingroup$ Use upper triangular matrices with non zero diagonals $\endgroup$
    – Alexandre Eremenko
    Commented Sep 5, 2013 at 22:05
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    $\begingroup$ @Alexandre Eremenko: What if the ring is not a division ring? Additionally, over a field with two elements, the identity matrix can only be written as a sum of two invertible matrices as follows: $\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix}1 & 1 \\ 1 & 0\end{pmatrix} + \begin{pmatrix}0 & 1 \\ 1 & 1\end{pmatrix}$ $\endgroup$
    – Ricardo Andrade
    Commented Sep 5, 2013 at 22:22
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    $\begingroup$ I agree that my proposal does not work for all rings. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 5, 2013 at 22:28
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    $\begingroup$ It is true for many rings. The terminology is somethig like "clean".There is also some USENET news posts on Sums of Invertible Matrices. A web search may help further. $\endgroup$
    – The Masked Avenger
    Commented Sep 6, 2013 at 1:06

2 Answers 2

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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

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This may help. Any $2\times 2$ matrix is a sum of four units:

$$ \begin{pmatrix} x & y \\ z & t \end{pmatrix} = \begin{pmatrix} x & 1 \\ -1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & -1 \\ 1 & t \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ z & -1 \end{pmatrix} + \begin{pmatrix} -1 & y \\ 0 & 1 \end{pmatrix}$$

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    $\begingroup$ If e.g. z and y are each the sum of two units, you can reduce this to two summands. $\endgroup$
    – The Masked Avenger
    Commented Sep 6, 2013 at 17:17
  • $\begingroup$ +1 For the nice identity. $\endgroup$
    – user38122
    Commented Sep 6, 2013 at 18:47

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