A ring is called a commutator ring if every element is a sum of additive commutators. In this note we give examples of such rings. In particular, we show that given any ring R , a right R -module N , and a nonempty set Ω, End R (⌖ Ω N ) and End R (Π Ω N ) are commutator rings if and only if either Ω is infinite or End R ( N ) is itself a commutator ring. We also prove that over any ring, a matrix having trace zero can be expressed as a sum of two commutators.
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Zachary Mesyan (2006) studied this question.