Abstract
Let R be a prime ring with nontrivial idempotents. We present a characterization of a tri-additive map f : R 3 → R such that f(x, y, z) = 0 for all x, y, z ∈ R with xy = yz = 0. As an application, we show that, in a prime ring with nontrivial idempotents, any local generalized (α, β)-derivation (or a generalized Jordan triple (α, β)-derivation) is a generalized (α, β)-derivation.
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