On the subring generated by commutators

Let [Formula: see text] be a ring. By the notation [Formula: see text] we denote the additive subgroup of [Formula: see text] generated by all [Formula: see text] in [Formula: see text]. In this work, we partially generalize a result due to Herstein [I. N. Herstein, Topics in Ring Theory (University of Chicago Press, 1969)] showing that if [Formula: see text], then the subring generated by [Formula: see text] is equal to [Formula: see text]. This result implies that [Formula: see text] cannot be a proper subring of [Formula: see text].