Commutators satisfying [A,B]^k = Id_n over C are characterized for pairs (k,n); in general unital rings, [a,b]^n=1 together with conditions on generated idempotents implies the ring is isomorphic to M_n(S) for some unital ring S.
Shoda, Einige S\"atze \"uber Matrizen, Jpn
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Commutators of finite multiplicative order
Commutators satisfying [A,B]^k = Id_n over C are characterized for pairs (k,n); in general unital rings, [a,b]^n=1 together with conditions on generated idempotents implies the ring is isomorphic to M_n(S) for some unital ring S.