Abstract

Let n be a fixed positive integer, R be a 2n!-torsion free prime ring and <TEX>$\mu$</TEX>, <TEX>$\nu$</TEX> be a pair of generalized derivations on R. If < <TEX>$\mu^2(x)+\nu(x),\;x^n$</TEX> > = 0 for all x <TEX>$\in$</TEX> R, then <TEX>$\mu$</TEX> and <TEX>$\nu$</TEX> 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 <TEX>$C_R$</TEX> and d, g be a pair of derivations on R. If < <TEX>$d^2(x)+g(x)$</TEX>, <TEX>$x^n$</TEX> > <TEX>$\in$</TEX> <TEX>$C_R$</TEX> for all x <TEX>$\in$</TEX> 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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