Derivations with values satisfying a right or left S_2n–polynomial

Abstract

By a right (left resp.) S_2n-polynomial we mean a multilinear polynomial f(X₁,…, X_t) over the ring of integers with noncommuting in-determinates X_isuch that for any prime ring R if f( X₁,…, X _t) is a PI of some nonzero right (left resp.) ideal of R, then R satisfies S_2nthe standard identity of degree 2n. In this paper we prove the theorem:Let R be a prime ring, d a nonzero derivation of R, L a noncommutative Lie ideal of R and f(X₁,…, X_t) a right or left S_2n-polynomial. Suppose that f(d( u₁)ⁿ¹,…,d(u_t)^nt)=0 for all u_iu,_i[d] L, where n₁,…,n_tare fixed positive integers. Then R satisfies S_2n+2. Also, the one-sided version of the theorem is given.