Commutators from a hyperplane of matrices

Abstract

Denote by M_n(K) the algebra of n by n matrices with entries in the field K. A theorem of Albert and Muckenhoupt states that every trace zero matrix of M_n(K) can be expressed as AB − BA for some pair (A, B) ∈ M_n(K)² . Assuming that n > 2 and that K has more than 3 elements, it is proved that the matrices A and B can be required to belong to an arbitrary given hyperplane of M_n(K).