B DHARA
Articles written in Proceedings – Mathematical Sciences
Volume 129 Issue 3 June 2019 Article ID 0040 Research Article
An identity on generalized derivations involving multilinear polynomials in prime rings
Let $R$ be a prime ring of characteristic different from $2$ with its Utumi ring of quotients $U$, extended centroid $C$, $f(x_{1},\ldots,x_{n})$ a multilinear polynomial over $C$, which is not central-valued on $R$ and $d$ a nonzero derivation of $R$. By $f(R)$, we mean the set of all evaluations of the polynomial $f(x_{1},\ldots,x_{n})$ in $R$. In the present paper, we study $b[d(u),u]+p[d(u),u]q+[d(u),u]c=0$ for all $u\in f(R)$, which includes left-sided, right-sided as well as two-sided annihilating conditions of the set $\{[d(u),u] : u\in f(R)\}$.We also examine some consequences of this result related to generalized derivations and we prove that if $F$ is a generalized derivation of $R$ and $d$ is a nonzero derivation of $R$ such that $$F^{2}([d(u), u])=0$$ for all $u\in f(R)$, then there exists $a\in U$ with $a^{2}=0$ such that $F(x)=xa$ for all $x\in R$ or $F(x)=ax$ for all $x\in R$.
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