Behavior like left Jordon multiplier map of $b$-generalized derivations in prime rings
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Suppose that $\R$ represents a prime ring of characteristic different from $2$ with its right Martindale quotient ring $\Q$, extended centroid $\C$ and $f(x_1,\ldots,x_n)$ symbolizes a noncentral multilinear polynomial over $\C$. Let $\I$ be a nonzero ideal of $\R$ and $b\in \Q$. If $\F$, $\G$ and $\h$ are three $b$-generalized derivations of $\R$ such that $$\F\big(f(s)\big)f(s)+\G\big(f(s)\big) f(s)v=\h\big(f(s)^2\big)$$ for all $s=(s_1,\ldots,s_n)\in \I^n$ and for some $v \in \R$, then we elaborate all probable constructions of the maps.
Abstract
Suppose that $\R$ represents a prime ring of characteristic different from $2$ with its right Martindale quotient ring $\Q$, extended centroid $\C$ and $f(x_1,\ldots,x_n)$ symbolizes a noncentral multilinear polynomial over $\C$. Let $\I$ be a nonzero ideal of $\R$ and $b\in \Q$. If $\F$, $\G$ and $\h$ are three $b$-generalized derivations of $\R$ such that $$\F\big(f(s)\big)f(s)+\G\big(f(s)\big) f(s)v=\h\big(f(s)^2\big)$$ for all $s=(s_1,\ldots,s_n)\in \I^n$ and for some $v \in \R$, then we elaborate all probable constructions of the maps.