Commutators of finite multiplicative order

Abstract

This article studies the equation $[A,B]^k = \Id_n$ for matrices over $\CC$,characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next generalized to matrix rings $M_n(S)$ over arbitrary unital rings $S$, where a sufficient condition on the unity of $S$ is established and explicit constructions of solutions are provided. Beyond matrix rings, the structural implications of the equation $[a,b{]}^n=1$ in a general unital ring $R$ are investigated, yielding a collection of idempotents whose properties govern the ring's structure. We prove that under a suitable condition on these idempotents, $[a,b{]}^n=1$ implies $R$ is isomorphic to $M_n(S)$ for some unital ring $S$. We also provide an alternative proof using a result on characterisation of matrix rings by Goyal and Khurana. These results together establish a framework connecting commutator equations and classical criteria for recognizing full matrix rings.

An alternative proof Theorem 4.9 has been added. All comments are welcome