Rings and C*-algebras generated by commutators

Abstract

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for matrix rings, and that one may choose $N \leq 3$ for rings that contain a direct sum of matrix rings -- this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that $N\leq 6$ can be arranged. For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. We prove that a C*-algebra cannot be a radical extension over a proper ideal, and we use this to deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.