Commutativity of rings involving additive mappings

Let R be an associative ring. In the present paper, we investigate commutativity of a ring admitting an additive mapping F satisfying any one of the following properties: (i) F ([x, y] 2 ) = F ([x 2 , y 2 ]), (ii) F ((x ◦ y) 2 ) = F (x 2 ◦ y 2 ), (iii) F ((xy) n ) = F (x n y n ), (iv) F (x m y n ) =...

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Bibliographic Details
Published in:Quaestiones mathematicae 2014-01, Vol.37 (2), p.215-229
Main Authors: Ali, Shakir, Ashraf, Mohammad, Khan, Mohammad Salahuddin, Vukman, Joso
Format: Article
Language:English
Subjects:
additive mapping
commuting mapping
derivation
generalized derivation
ideal
Prime ring
semiprime ring
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Summary:Let R be an associative ring. In the present paper, we investigate commutativity of a ring admitting an additive mapping F satisfying any one of the following properties: (i) F ([x, y] 2 ) = F ([x 2 , y 2 ]), (ii) F ((x ◦ y) 2 ) = F (x 2 ◦ y 2 ), (iii) F ((xy) n ) = F (x n y n ), (iv) F (x m y n ) = F (y n x m ), (v) (F (x)F (y)) n = (F (y)F (x)) n for all x, y ∈ R, where m and n are positive integers greater than 1. Moreover, some related results are also discussed. Finally, some examples are given to demonstrate that the restrictions imposed on the hypotheses of the various results are not superfluous.
ISSN:1607-3606
1727-933X
DOI:10.2989/16073606.2013.779994