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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6$\begingroup$ The body of the question is unclear and doesn't correspond to the title. $\endgroup$– Victor ProtsakCommented Aug 23, 2010 at 19:26
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$\begingroup$ 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". $\endgroup$– Pietro MajerCommented Aug 24, 2010 at 19:13
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1 Answer
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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.
answered Aug 23, 2010 at 20:11
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1$\begingroup$ 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 $\endgroup$ Commented Aug 24, 2010 at 11:54
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$\begingroup$ 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. $\endgroup$ Commented Aug 24, 2010 at 17:28
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$\begingroup$ 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 $\endgroup$ Commented Aug 24, 2010 at 17:58