Sum of two unitary matrix is equal to every matrix? [closed]

Let $R=M_{n}(Z_{2})$, can we write every matrices of $R$ as sum of two matrices of $GL_{n}(Z_{2})$?

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The body of the question is unclear and doesn't correspond to the title. –  Victor Protsak Aug 23 '10 at 19:26
However, if a question like this appears unclear, I would rather ask the questioner to clarify it. In the present case, the only unclear point seems to me the use of "unitary" in the title, in place of "invertible". –  Pietro Majer Aug 24 '10 at 19:13
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closed as not a real question by Victor Protsak, Andrew Stacey, Akhil Mathew, Captain Oates, Qiaochu YuanAug 23 '10 at 21:20

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

1 Answer

It appears you are asking whether $R$ is a 2-good ring. The answer is yes. You may find the paper "2-good rings" by Peter Vamos to be useful in giving you some background information.

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The answer is "yes" if all matrices are the sum of at most 2 units. The answer is "almost" if every matrix is required to be the sum of exactly two units. (Hint: the monkey in this wrench is pretty small.) Gerhard "Ask Me About System Design" Paseman, 2010.08.24 –  Gerhard Paseman Aug 24 '10 at 11:54
I failed to remark that I am, of course, assuming that $n>1$, otherwise the result is not true. I also want to mention that a colleague of mine Thomas Dorsey (along with two coauthors) has a preprint on this topic which is quite good. You might look for that paper to appear soon. –  Pace Nielsen Aug 24 '10 at 17:28
Thanks for the heads up on the preprint. If you see the paper before I do, please mention it here. I shall do same if I see the paper before a new comment here. Gerhard "Ask Me About System Design" Paseman, 2010.08.24 –  Gerhard Paseman Aug 24 '10 at 17:58
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