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In one paper from 1980 I found a note that there are no known algorithms for solving homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring. (The non-commutative ring is the ring which elemens are linear operators with addition and composition operations.)

Are there any algorithms for solving such homogenous equations?

Is there any progress in this direction?

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    $\begingroup$ In 1980 there were already known examples of countable (non-commutative, associative) rings with unsolvable word problem, and probably that can be chosen to have no zero divisors. For such rings those matrix equations are not solvable even for $1\times 1$-matrices $\endgroup$
    – YCor
    Apr 29, 2014 at 10:14
  • $\begingroup$ Thank you, Yves. Do you know is there characterization of the countable non-comm rings for which word problem is unsolvable? $\endgroup$ Apr 29, 2014 at 17:50
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    $\begingroup$ In general I don't expect any useful characterization except obvious restatements (and maybe embedding theorems, which won't help much). I was especially thinking of group algebras $\mathbf{Z}[G]$ of finitely generated groups $G$: such an algebra has a solvable word problem if and only $G$ has a solvable word problem. $\endgroup$
    – YCor
    Apr 29, 2014 at 21:34

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