### Joso Vukman

Department of Mathematics, University of Maribor, PEF, Koroška 160, 2000 Maribor, Slovenia
e-mail: joso.vukman@uni-mb.si

Abstract.   In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be integers and let R be a 2mn(m+n-1)!-torsion free semiprime ring. Suppose there exist derivations D, G : R R such that D(xm)xn + xnG(xm) = 0 holds for all x R. In this case both derivations D and G map R into its center and D = -G. We apply this purely algebraic result to obtain a range inclusion result of continuous derivations on Banach algebras.

2000 Mathematics Subject Classification.   16N60, 46K15.

Key words and phrases.   Prime ring, semiprime ring, Banach algebra, derivation, commuting mapping, centralizing mapping, skew-commuting mapping, skew-centralizing mapping.

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DOI: 10.3336/gm.40.2.01

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