On 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 $[a_1,a_2,\dots ,a_k]$ $(a_i\in A)$ where $[a_1,a_2]=a_1{a}_2-a_2{a}_1$, $[a_1,\dots ,a_{k-1},a_k]=[[a_1,\dots ,a_{k-1}],a_k]$ $(k>2)$. It has been known that, if either $m$ or $n$ is odd then \[ 6 \, [a_1, a_2, \dots , a_m] [b_1, b_2, \dots , b_n] \in T^{(m+n-1)} \] for all $a_i,b_j\in A$. 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 $a_i,b_j\in A$, \[ 3 \, [a_1, a_2, \dots , a_m] [b_1, b_2, \dots , b_n] \in T^{(m+n-1)}. \] Since it has been known that, in general, \[ [a_1, a_2, a_3] [b_1, b_2] \notin T^{(4)}, \] our result cannot be improved further for all $m,n$ such that at least one of them is odd.

7 pages