Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let φ(x, y) = [x, y] t -[x, y] m [α([x, y]),[x, y]]n [x, y] s . Thus, (a) if φ(u, v) = 0 for any u, v L, then L ⊂ Z(R); (b) if φ(u, v) Z(R) for any u, v L, then either L ⊂ Z(R) or R satisfies s 4 , the standard identity of degree 4. We also extend the results to semiprime rings. © 2019 Academy of Mathematics and Systems Science, Chinese Academy of Sciences.
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