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Consider $M_2(\mathbb{Z})$. Is it possible to find two matrices A,B such that their commutator AB - BA equals a given matrix C? Is there any chance to characterize all possible occuring commutators in a given set of matrices over some fixed ring? One specific example: C the unit matrix. Are there such A,B? To find them i would try to solve a system of equations. But is there are more systematic way to say theoretically that we cannot or can find A,B, such that AB-BA = 1?

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up vote 4 down vote accepted

For the specific case of $[A,B]=I$, the answer is No over most rings - check the trace!

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