Pair of (generalized-)derivations on rings and Banach algebras
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Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and μ, be a pair of generalized derivations on R. If (μ2(x)+ v(x), x^n = 0 for all x ∈ R, then μ and are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!- torsion free prime ring with the center CR and d, g be a pair of derivations on R. If (d^2(x) + g(x), x^n)∈ CR for all x ∈ R, then d = g = 0. Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.
Abstract
Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and μ, be a pair of generalized derivations on R. If (μ2(x)+ v(x), x^n = 0 for all x ∈ R, then μ and are either left multipliers or right multipliers. Let n be a fixed positive integer, R be a noncommutative 2n!- torsion free prime ring with the center CR and d, g be a pair of derivations on R. If (d^2(x) + g(x), x^n)∈ CR for all x ∈ R, then d = g = 0. Then we apply these purely algebraic techniques to obtain several range inclusion results of pair of (generalized-)derivations on a Banach algebra.
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