An identity with derivations on rings and Banach algebrasFošner, Ajda (Avtor) Fošner, Maja (Avtor) Vukman, Joso (Avtor) mathematicsalgebraassociative rings and algebrasprime ringsBanach algebrasidentitiesderivationsThe main purpose of this paper is to study the following: Let m, n, and $k_{i}, i = 1, 2, ..., n$ be positive integers and let $R$ be a $2m(m+ k_{1} + k_{2} + ... + k_{n} -1)!$-torsion free semiprime ring. Suppose that there exist derivations $D_{i} : R \to R, i = 1, 2, ..., n + 1$ , such that $D_{1}(x^{m})x^{k_{1}+...+k_{n}}+x^{k_{1}} D_{2}(x^{m})x^{k_{2}+...+k_{n}}+...+x^{k_{1}+...+k_{n}}D_{n+1}(x^{m})=0$ holds for all $x \in R$. Then we prove that $D_{1}+D_{2}+...+D_{n+1}=0$ and that the derivation $k_{1}D_{2}+(k_{1}+k_{2})D_{3}+...+(k_{1}+k_{2}+...+k{n})D_{n+1}$ maps $R$ into its center. We also obtain a range inclusion result of continuous derivations on Banach algebras.De Gruyter Brill20082017-03-31 09:17:35Znanstveno delo65327sl