In this talk, we study the equation [𝐴,𝐵]𝑘 = Id𝑛 for matrices over C. In fact, we characterize the pairs (𝑘, 𝑛) for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. We then discuss the same problem over the generalized matrix rings 𝑀𝑛(𝑆), 𝑆 being an arbitrary unital ring. We provide a sufficient condition on the unity of 𝑆 and construct explicit solutions. Further, we look beyond matrix rings and investigate the structural implications of the equation [𝑎, 𝑏]𝑛 = 1 in a general unital ring 𝑅 by yielding a collection of idempotents. We prove that under a suitable condition on these idempotents, [𝑎, 𝑏]𝑛 = 1 implies 𝑅 is isomorphic to 𝑀𝑛(𝑆) for some unital ring 𝑆.
These results establish a framework connecting commutator equations and classical criteria for recognizing full matrix rings. The talk is based on a joint work with Gobinda Sau and Arindam Sutradhar, (cf. https://arxiv.org/abs/2605.09451).
