Generalized commutators in matrix rings

Abstract

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 × 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 × n traceless matrices are necessarily commutators over a commutative base ring, for any n ≥ 2.

Keywords:

• matrix rings
• commutators
• generalized commutators
• traceless matrices
• elementary divisor rings

AMS Subject Classifications::

• 15A23
• 15B33
• 16S50
• 47B47
• 15A27