# Article

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Keywords:
prime ring; derivation; generalized derivation; extended centroid; Utumi quotient ring
Summary:
Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$, let $F$, $G$ and $H$ be three generalized derivations of $R$, $I$ an ideal of $R$ and $f(x_1,\ldots ,x_n)$ a multilinear polynomial over $C$ which is not central valued on $R$. If $$F(f(r))G(f(r))=H(f(r)^2)$$ for all $r=(r_1,\ldots ,r_n) \in I^n$, then one of the following conditions holds: \item {(1)} there exist $a\in C$ and $b\in U$ such that $F(x)=ax$, $G(x)=xb$ and $H(x)=xab$ for all $x\in R$; \item {(2)} there exist $a, b\in U$ such that $F(x)=xa$, $G(x)=bx$ and $H(x)=abx$ for all $x\in R$, with $ab\in C$; \item {(3)} there exist $b\in C$ and $a\in U$ such that $F(x)=ax$, $G(x)=bx$ and $H(x)=abx$ for all $x\in R$; \item {(4)} $f(x_1,\ldots ,x_n)^2$ is central valued on $R$ and one of the following conditions holds: \itemitem {(a)} there exist $a,b,p,p'\in U$ such that $F(x)=ax$, $G(x)=xb$ and $H(x)=px+xp'$ for all $x\in R$, with $ab=p+p'$; \itemitem {(b)} there exist $a,b,p,p'\in U$ such that $F(x)=xa$, $G(x)=bx$ and $H(x)=px+xp'$ for all $x\in R$, with $p+p'=ab\in C$.
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