Products of commutators in certain rings

Abstract

Let [Formula: see text] be a ring. For each positive integer [Formula: see text], [Formula: see text] (respectively, [Formula: see text]) is denoted by the set of all products of (respectively, at most) [Formula: see text] additive commutators of [Formula: see text]. If [Formula: see text] is an algebra over a field [Formula: see text] such that [Formula: see text] for some [Formula: see text], then [Formula: see text]. If [Formula: see text] is a noncommutative ring such that [Formula: see text] for some [Formula: see text], then the necessary and sufficient condition for [Formula: see text] to be a division ring is that every nonzero element of [Formula: see text] is invertible. If [Formula: see text] is the matrix ring over a division ring [Formula: see text], then [Formula: see text], and in particular, if [Formula: see text] is a field, then [Formula: see text].