Commutator rings

Abstract

A ring is called a commutator ring if every element is a sum of additive commutators. In this paper we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a set X, End_R(\bigoplus_X N) and End_R(\prod_X N) are commutator rings if and only if either X 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.

9 pages. The final version contains two new results: Propositions 14 and 20. Also, some of the proofs have been rewritten to improve clarity

Cited by in corpus (6)

• Word equations in simple groups and polynomial equations in simple algebras
• Polynomials of small degree evaluated on matrices
• Simple Lie algebras arising from Leavitt path algebras
• Commutator Leavitt path algebras
• Traces on Semigroup Rings and Leavitt Path Algebras
• Bracket width of current Lie algebras