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Sibirsk. Mat. Zh., 2017, Volume 58, Number 1, Pages 3–15 (Mi smj2834)  

Centralizers of generalized skew derivations on multilinear polynomials

E. Albaşa, N. Argaça, V. De Filippisb

a Department of Mathematics, Science Faculty, Ege University, Bornova, Izmir, Turkey
b M.I.F.T., University of Messina, Italy

Abstract: Let $\mathscr R$ be a prime ring of characteristic different from $2$, let $\mathscr Q$ be the right Martindale quotient ring of $\mathscr R$, and let $\mathscr C$ be the extended centroid of $\mathscr R$. Suppose that $\mathscr G$ is a nonzero generalized skew derivation of $\mathscr R$ and $f(x_1,…,x_n)$ is a noncentral multilinear polynomial over $\mathscr C$ with $n$ noncommuting variables. Let $f(\mathscr R)=\{f(r_1,…,r_n)\colon r_i\in\mathscr R\}$ be the set of all evaluations of $f(x_1,…,x_n)$ in $\mathscr R$, while $\mathscr A=\{[\mathscr G(f(r_1,…,r_n)),f(r_1,…,r_n)]\colon r_i\in\mathscr R\}$, and let $C_\mathscr R(\mathscr A)$ be the centralizer of $\mathscr A$ in $\mathscr R$; i.e., $C_\mathscr R(\mathscr A)=\{a\in\mathscr R\colon[a,x]=0 \forall x\in\mathscr A\}$. We prove that if $\mathscr A\neq(0)$, then $C_\mathscr R(\mathscr A)=Z(R)$.

Keywords: polynomial identity, generalized skew derivation, prime ring.

DOI: https://doi.org/10.17377/smzh.2017.58.101

Full text: PDF file (341 kB)
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English version:
Siberian Mathematical Journal, 2017, 58:1, 1–10

Bibliographic databases:

UDC: 512.552
MSC: 16W25, 16N60
Received: 11.05.2015

Citation: E. Albaş, N. Argaç, V. De Filippis, “Centralizers of generalized skew derivations on multilinear polynomials”, Sibirsk. Mat. Zh., 58:1 (2017), 3–15; Siberian Math. J., 58:1 (2017), 1–10

Citation in format AMSBIB
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\paper Centralizers of generalized skew derivations on multilinear polynomials
\jour Sibirsk. Mat. Zh.
\yr 2017
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\issue 1
\pages 3--15
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\crossref{https://doi.org/10.17377/smzh.2017.58.101}
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\transl
\jour Siberian Math. J.
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\vol 58
\issue 1
\pages 1--10
\crossref{https://doi.org/10.1134/S0037446617010013}
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