Generalized commutators in matrix rings

An element of the form [a, b, c] = abc - cba in a ring is called a generalized commutator. In this article, we show that, in a matrix ring M-n(S) (n >= 2) over any ring S (with identity), every matrix is a sum of a commutator and a generalized commutator. If S is an elementary divisor ring (in the sense that every square matrix over S is equivalent to a diagonal matrix), then every matrix in M-n(S) (n >= 2) is, in fact, a generalized commutator. Using suitable generic techniques, we show, however, that this conclusion is not true in general for various rings S (e.g. polynomial rings and power series rings). Indeed, for any n >= 2, there exists an n x n traceless matrix over some commutative ring S that is not a generalized commutator (respectively, is a generalized commutator but not a commutator) in M-n(S). This gives, in particular, a strong negative solution to the problem whether n x n traceless matrices are necessarily commutators over a commutative base ring, for any n >= 2.