Some products of commutators in an associative ring

Abstract

Let [Formula: see text] be a unital associative ring and let [Formula: see text] be the two-sided ideal of [Formula: see text] generated by all commutators [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] [Formula: see text]. It has been known that, if either [Formula: see text] or [Formula: see text] is odd, then [Formula: see text] for all [Formula: see text]. This was proved by Sharma and Srivastava in 1990 and independently rediscovered later (with different proofs) by various authors. The aim of our paper is to give a simple proof of the following result: if at least one of the integers [Formula: see text] is odd, then for all [Formula: see text], [Formula: see text] Since it has been known that, in general, [Formula: see text] our result cannot be improved further for all [Formula: see text] such that at least one of them is odd.