We first give several polynomial identities of semiprime rings which make the additive mappings appearing in the identities to be generalized derivations. Then we study pairs of generalized Jordan derivations on prime rings. Let m,n be fixed positive integers,R be a noncommutative 2(m+n)!-torsion free prime ring with the center Z and μ, ν be a pair of generalized Jordan derivations on _R_R. Ifμ(x__m)x__n+_x__n_ν(x__m) ∈ Z for all x ∈ R, then μ and ν are left (or right) multipliers.In particular, if μ, ν are a pair of derivations onR satisfying the same assumption, then μ = ν = 0. Then applying these purely algebraic result we obtain several range inclusion results of pair of derivations on Banach algebras.
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Feng Wei, Zhankui Xiao, Generalized Derivations on (Semi-)Prime Rings and Noncommutative Banach Algebras. Rend. Sem. Mat. Univ. Padova 122 (2009), pp. 171–190