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Title:
Special type of additive maps in prime rings with annihilating and centralizing condition
PDFSource:
Discussiones Mathematicae - General Algebra and Applications
Received: 2024-03-22 , Revised: 2024-11-20 , Accepted: 2024-11-20 , Available online: 2025-08-13 , https://doi.org/10.7151/dmgaa.1486
Abstract:
Let $R$ be a prime ring with char $R\neq 2$ and $f(r_1,\dots,r_n)$ be a non-central
multilinear polynomial over $C(=Z(U))$, where $U$ is the Utumi ring of quotients
of $R$. Let $I$ be a nonzero two sided ideal of $R$, $L$ a non central Lie ideal
of $R$ and $\mathscr{F}$, $\mathscr{G}$ two generalized derivations of $R$. Denote the set
$f(I)=\{f(r_1,\ldots,r_n) | r_1,\dots,r_n\in I\}$. If for some $0\neq a\in R$,
$$
a[(\mathscr{F}^2+\mathscr{G})(u), u]\in C
$$
for all $u\in f(I)$ or $u\in L$, then possible forms of the maps are described.
This result improves the result proved by De Filippis et al. in [8]
and Carini and Scudo in [6].
Keywords:
prime ring, derivation, generalized derivation
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