Some products of commutators in an associative ring

Abstract

Let A be a unital associative ring and let T (k) be the two-sided ideal of A generated by all commutators [a1, a2, . . . , a k ] (ai ∈ A) where [a1, a2] = a1a2−a2a1, [a1, . . . , a k−1 , a k ] = [a1, . . . , a k−1 ], a k (k > 2). It has been known that, if either m or n is odd thenfor all ai, bj ∈ A. This was proved by Sharma and Srivastava in 1990 and independently rediscovered later (with different proofs) by various authors. The aim of our note is to give a simple proof of the following result: if at least one of the integers m, n is odd then, for all ai, bj ∈ A, 3 [a1, a2, . . . , am][b1, b2, . . . , bn] ∈ T (m+n−1) .Since it has been known that, in general, [a1, a2, a3][b1, b2] / ∈ T (4) , our result cannot be improved further for all m, n such that at least one of them is odd. AMS MSC Classification: 16R10, 16R40

Relatively free associative Lie nilpotent algebras of rank 3

“…[12], [13], [7]). Наиболее общий результат в этом направлении получен недавно в [14]. В работе [7] это утверждение называлось теоремой о произведении T (m) T (n) .…”