Commutators of finite multiplicative order

TL;DR

This paper investigates solutions to the commutator equation $[A,B]^k = Id$ over complex matrices and general rings, linking solutions to ring structure and classical roots of unity results.

Contribution

It generalizes the commutator equation to arbitrary unital rings, providing explicit constructions and structural implications that connect to matrix ring recognition.

Findings

Characterization of $(k,n)$ pairs with solutions over $\, ext{C}$.

Explicit constructions of solutions in matrix rings over unital rings.

Structural conditions under which rings are isomorphic to matrix rings.

Abstract

This article studies the equation $[A,B]^k = {\rm Id}_n$ for matrices over $\mathbb{C}$, 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 $1_S$ 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 $R \cong M_n(S)$ for some unital ring $S$. These results together establish a framework connecting commutator equations and classical criteria for recognizing full matrix rings.