Characterization of higher commutators

Let R be a ring having the property that every proper ideal of R is contained in a maximal ideal of R (in particular, if R is finitely generated as an ideal). Generalizing several known results, we characterize higher commutators V of R whenever R is generated by V (respectively, [R,V]) as an ideal. In particular, if V is a higher commutator of a unital ring R with 1∈V, then V is equal to either R or [R,R], or [[R,R],[R,R]]. Given a semiprime ring R, which is generated by [R,R], all higher commutators of R are obtained if R possesses a central higher commutator. We also characterize all higher commutators of Mn(D) for n≥2 when D is a unital commutative ring. In addition, if D is 2-torsion free and 2D⊊D, then M2(D) has infinitely many higher commutators. © 2022 Elsevier Inc.