Let \(R\) be a ring which satisfies the identity \((x,y,z) = (y,z,x)\) and is without elements of orders 2 and 3 in its additive group. The author proves that \(A^ 2 = 0\), where \(A\) is the alternator ideal of \(R\). In particular, let \(R\) also be a nil ring of bounded index \(n\). Then using results due to Shirshov and Zhevlakov, \(R\) is locally nilpotent; and if \(R\) does not have elements of order \(\leq n\) in its additive group, then \(R\) is solvable of index \(\leq {n(n + 1) \over 2} + 1\).