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## Annales Mathematicae Silesianae

2019 | 33 | 266--275
Tytuł artykułu

### A Note on Multiplicative (generalized) (α,β)-Derivations in Prime Rings

Autorzy
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let $R$ be a prime ring with center $Z(R)$. A map $G\colon R\to R$ is called a multiplicative (generalized) $(\alpha,\beta)$-derivation if $G(xy) = G(x)\alpha(y)+\beta(x)g(y)$ is fulfilled for all $x,y\in R$, where $g\colon R\to R$ is any map (not necessarily derivation) and $\alpha,\beta\colon R\to R$ are automorphisms. Suppose that $G$ and $H$ are two multiplicative (generalized) $(\alpha,\beta)$-derivations associated with the mappings $g$ and $h$, respectively, on $R$ and $\alpha,\beta$ are automorphisms of $R$. The main objective of the present paper is to investigate the following algebraic identities: (i) $G(xy) + \alpha(xy) = 0$, (ii) $G(xy) + \alpha(yx) = 0$, (iii) $G(xy) + G(x)G(y) = 0$, (iv) $G(xy) = \alpha(y) \circ H(x)$ and (v) $G(xy) = [\alpha(y),H(x)]$ for all $x,y$ in an appropriate subset of $R$.(original abstract)
Słowa kluczowe
PL
EN
Czasopismo
Rocznik
Tom
Strony
266--275
Opis fizyczny
Twórcy
autor
• Aligarh muslim University Department of Mathematics
autor
• Ibb University Department of Mathematics ,
• King Abdulaziz University Department of Mathematics, Faculty of Science
Bibliografia
• E. Albaş, Generalized derivations on ideals of prime rings, Miskolc Math. Notes 14 (2013), no. 1, 3-9.
• M. Ashraf, A. Ali and S. Ali, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math. 31 (2007), no. 3, 415-421.
• M. Ashraf and N. Rehman, On derivation and commutativity in prime rings, East-West J. Math. 3 (2001), no. 1, 87-91.
• M. Brešar, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991), no. 1, 89-93.
• M.N. Daif, When is a multiplicative derivation additive?, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 615-618.
• M.N. Daif and M.S. Tammam El-Sayiad, Multiplicative generalized derivations which are additive, East-West J. Math. 9 (2007), no. 1, 31-37.
• B. Dhara and S. Ali, On multiplicative (generalized)-derivations in prime and semiprime rings, Aequationes Math. 86 (2013), no. 1-2, 65-79.
• H. Goldmann and P. Šemrl, Multiplicative derivation on C(X), Monatsh. Math. 121 (1996), no. 3, 189-197.
• B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), no. 4, 1147-1166.
• W.S. Martindale, III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21 (1969), no. 3, 695-698.
• E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), no. 6, 1093-1100.
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