<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>R</italic> be a prime ring with center <italic>Z</italic>(<italic>R</italic>). A map <italic>G</italic> : <italic>R</italic> →<italic>R</italic> is called a multiplicative (generalized) (<italic>α</italic>, <italic>β</italic>)-derivation if <italic>G</italic>(<italic>xy</italic>)= <italic>G</italic>(<italic>x</italic>)<italic>α</italic>(<italic>y</italic>)+<italic>β</italic>(<italic>x</italic>)<italic>g</italic>(<italic>y</italic>) is fulfilled for all <italic>x</italic>; <italic>y</italic> ∈ <italic>R</italic>, where <italic>g</italic> : <italic>R</italic> → <italic>R</italic> is any map (not necessarily derivation) and <italic>α</italic>; <italic>β</italic> : <italic>R</italic> → <italic>R</italic> are automorphisms. Suppose that <italic>G</italic> and <italic>H</italic> are two multiplicative (generalized) (<italic>α</italic>, <italic>β</italic>)-derivations associated with the mappings <italic>g</italic> and <italic>h</italic>, respectively, on <italic>R</italic> and <italic>α</italic>, <italic>β</italic> are automorphisms of <italic>R</italic>. The main objective of the present paper is to investigate the following algebraic identities: (<italic>i</italic>) <italic>G</italic>(<italic>xy</italic>) + <italic>α</italic>(<italic>xy</italic>) = 0, (<italic>ii</italic>) <italic>G</italic>(<italic>xy</italic>) + <italic>α</italic>(<italic>yx</italic>) = 0, (<italic>iii</italic>) <italic>G</italic>(<italic>xy</italic>) + <italic>G</italic>(<italic>x</italic>)<italic>G</italic>(<italic>y</italic>) = 0, (<italic>iv</italic>) <italic>G</italic>(<italic>xy</italic>) = <italic>α</italic>(<italic>y</italic>) ○ <italic>H</italic>(<italic>x</italic>) and (<italic>v</italic>) <italic>G</italic>(<italic>xy</italic>) = [<italic>α</italic>(<italic>y</italic>), <italic>H</italic>(<italic>x</italic>)] for all <italic>x</italic>, <italic>y</italic> in an appropriate subset of <italic>R</italic>.</p></abstract>

Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.

A Note on Multiplicative (Generalized) (<italic>α</italic>, <italic>β</italic>)-Derivations in Prime Rings

Department of Mathematics, Faculty of science

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

A Note on Multiplicative (Generalized) (, )-Derivations in Prime Rings

Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for...

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

Publié en ligne: 18 juil. 2019

Pages: 266 - 275

Reçu: 01 juil. 2018

Accepté: 03 avr. 2019

DOI: https://doi.org/10.2478/amsil-2019-0008

Mots clés
prime ring, multiplicative (generalized)(, )-derivation, left ideal

© 2019 Nadeem ur Rehman et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Publié en ligne: 18 juil. 2019

Pages: 266 - 275

Reçu: 01 juil. 2018

Accepté: 03 avr. 2019

DOI: https://doi.org/10.2478/amsil-2019-0008

Mots clésprime ring, multiplicative (generalized)(, )-derivation, left ideal

© 2019 Nadeem ur Rehman et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
prime ring, multiplicative (generalized)(, )-derivation, left ideal