On higher commutators of rings

Abstract

For a subset T of a ring R we denote by I (T) the ideal of R generated by T. Given a higher commutator L of R, if R = I (L) then R = L + L2? The question is motivated by the result that a ring R is equal to its subring generated by [R,R] if R is either a noncommutative simple ring (by Herstein) or a unital ring with 1 [R,R] (by Eroǧlu). In this note, we study the question for the rings R satisfying the property that every proper ideal of R is contained in a maximal ideal (in particular, if R is finitely generated as an ideal).