Article in volume
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Title:
Additive mappings satisfying algebraic identities in semiprime rings
PDFSource:
Discussiones Mathematicae - General Algebra and Applications 43(2) (2023) 327-337
Received: 2021-08-29 , Revised: 2022-05-23 , Accepted: 2022-05-23 , Available online: 2023-01-13 , https://doi.org/10.7151/dmgaa.1422
Abstract:
Let $R$ be a $k$-torsion free semiprime ring. Suppose that $F, d : R\to R$ be
two additive mappings which satisfy the algebraic identity $F(x^{2n})=F(x^n)
\alpha(x^n)+ \beta(x^n) d(x^n)$ for all $x\in R$, where $\alpha$ and $\beta$
are automorphisms on $R$. Then $F$ is a generalized $(\alpha,\beta)$-derivation
with associated $(\alpha,\beta)$-derivation $d$ on $R$, where $k\in\{2,n,2n-1\}$.
On the other hand, it is proved that $f$ is a generalized Jordan left
$(\alpha, \beta)$-derivation associated with Jordan left
$(\alpha, \beta)$-derivation $\delta$ on $R$ if they satisfy the algebraic
identity $f(x^{2n})=\alpha(x^n) f(x^n)+ \beta(x^n)\delta(x^n)$ for all $x\in R$
together with some restrictions on $R$.
Keywords:
semiprime rings, generalized $(\alpha, \beta)$-derivation, generalized left $(\alpha, \beta)$-derivation and additive mappings
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