ON RINGS WITH ENGEL CYCLES

A ring R is called an EC-ring if for each x, y is a member of R, there exist distinct positive integers m, n such that the extended commutators [x, y]m and [x, y]n are equal. We show that in certain EC-rings, the commutator ideal is periodic; we establish commutativity of arbitrary EC-domains; we prove that a ring R is commutative if for each x, y is a member of R, there exists n > 1 for which [x, y] = [x, y]n.