Let R be a ring, and N the set of all nilpotent elements of R. We prove thefollowingTheorem if a rinq R satisfies the following condition: for all x1…,xk in R, there existwords ω2 = ω1(x2,…,xk ) and ω2 = ω2(x1…,xk-1 depending on x1 …,xk such that|ω1|xk> 1,|ω2|>1, and|ω1|≠|ω2|. Suppose that x1 ω1 (x2…,xk) = ω2( x1…,xk-1)xk where k >1 is a fixed integer. Then(1) N jorms an ideal of R,(2) R = N + R1, where R1 is isomorphic to a subdirect sum of fields. In particular, if N is commutative, then R is commutative. |