Bulletin of the
Korean Mathematical Society BKMS

Let $n$ be a fixed positive integer, $\mathcal{R}$ be a $2n!$-torsion free prime ring and $\mu, \nu$ be a pair of generalized derivations on $\mathcal{R}$. If $\langle\mu^2(x)+\nu(x),x^n\rangle=0$ for all $x\in \mathcal{R}$, then $\mu$ and $\nu$ are either left multipliers or right multipliers. Let $n$ be a fixed positive integer, $\mathcal{R}$ be a noncommutative $2n!$-torsion free prime ring with the center $\mathcal{C_R}$ and $d, g$ be a pair of derivations on $\mathcal{R}$. If $\langle d^2(x)+g(x), x^n\rangle\in \mathcal{C_R}$ for all $x\in \mathcal{R}$, 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.

Keywords: (generalized-)derivation, (semi-)prime ring, Banach algebra

MSC numbers: 16W25, 16N60, 47B47