2026/05/31 by Arijit Mukherjee, Gobinda Sau, Arindam Sutradhar
Mathematics · #math.RA #msc:15A15 #msc:16U40 #msc:16U60
An alternative proof Theorem 4.9 has been added. All comments are welcome
arxiv created 2026/07/30 · arxiv updated 2026/07/31
This article studies the equation [A,B]k = \Idn 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 Mn(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 Mn(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.