A note on certain central differential identities with generalized derivations

Let R be a noncommutative prime ring of characteristic di erent from 2 with right Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let $f(x_1,...,x_n)$ be a non-central multilinear poly- nomial over C, $m geq 1$ a fi xed integer,a a xed element of R,G a non-zero generalized derivation of R. If $aG(f(r_1,....,r_n))^m in Z(R)$ for all $r_1,....,r_n in I$, then one of the following holds:(1) aI=aG(I) = (0); (2) G(x)=qx, for some $q in U$ and aqI=0; (3) [f(x_1,...,x_n),x_{n+1}]x_{n+2}$ is an identity for I; (4) G(x)=cx + [q;x] for all $x in R$, where $c;q in U$ such that cI = 0 and [q;I]I= 0; (5) $dim_C(RC)leq 4$; (6) $G(x) =alphax$, for some $alpha in C$; moreover a 2 C and $f(x_1,...,x_n)^m$ is central valued on R.

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