Abstract
Let R be a noncommutative semi prime ring. Suppose that there exists a derivation d : R <TEX>$\to$</TEX> R such that for all x <TEX>$\in$</TEX> R, either [[d(x),x], d(x)] = 0 or <TEX>$\langle$</TEX><TEX>$\langle(x),\;x\rangle,\;d(x)\rangle$</TEX> = 0. In this case [d(x), x] is nilpotent for all x <TEX>$\in$</TEX> R. We also apply the above results to a Banach algebra theory.
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