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Fošner [4] defined a generalized module left (m, n)-derivation and proved the Hyers-Ulam stability of generalized module left (m, n)-derivations. In this note, we prove that every generalized module left (m, n)-derivation is trival if the algebra is unital and m 6= n. 1. Stability of Module Left (m,n)-derivations Let A be an algebra and M be a left A-module. An additive mapping d : A → M is called a module left derivation if d(xy) = x · d(y) + y · d(x) for all x, y ∈ A. Definition 1.1 ([3]). Let A be an algebra and M be a left A-module. An additive mapping d : A → M is called a module left (m, n)-derivation if (m + n)d(xy) = 2mx · d(y) + 2ny · d(x) (1.1) for all x, y ∈ A. Here m and n are nonnegative integers with m + n 6= 0. Let A be an algebra and M be a left A-module. An additive mapping g : A → M is called a generalized module left derivation if there exists a module left derivation d : A → M such that g(xy) = x · g(y) + y · d(x) for all x, y ∈ A. Definition 1.2 ([4]). Let A be an algebra and M be a left A-module. An additive mapping g : A → M is called a generalized module left (m,n)-derivation if there exists a module left (m,n)-derivation d : A → M such that (m + n)g(xy) = 2mx · g(y) + 2ny · d(x) (1.2) for all x, y ∈ A. Here m and n are nonnegative integers with m + n 6= 0. Received by the editors October 17, 2016. Accepted October 24, 2016. 2010 Mathematics Subject Classification. 16W25, 16D60, 39B62.

GENERALIZED MODULE LEFT (m, n)-DERIVATIONS