Multiplicative (generalized)-derivations and left ideals in semiprime rings

Authors : Asma Ali, Basudeb Dhara, Shahoor Khan

Pages : 1293-1306

View : 18 | Download : 7

Publication Date : 2015-12-01

Article Type : Research

Abstract :Let R be a semiprime ring with center Z ( R ) . A mapping F : R → R (not necessarily additive) is said to be a multiplicative (generalized)- derivation if there exists a map f : R → R (not necessarily a derivation nor an additive map) such that F ( xy ) = F ( x ) y + xf ( y ) holds for all x,y ∈ R . The objective of the present paper is to study the following identities: (i) F ( x ) F ( y ) ± [ x,y ] ∈ Z ( R ) , (ii) F ( x ) F ( y ) ± x ◦ y ∈ Z ( R ) , (iii) F ([ x,y ]) ± [ x,y ] ∈ Z ( R ) , (iv) F ( x ◦ y ) ± ( x ◦ y ) ∈ Z ( R ) , (v) F ([ x,y ]) ± [ F ( x ) ,y ] ∈ Z ( R ) , (vi) F ( x ◦ y ) ± ( F ( x ) ◦ y ) ∈ Z ( R ) , (vii) [ F ( x ) ,y ] ± [ G ( y ) ,x ] ∈ Z ( R ) , (viii) F ([ x,y ]) ± [ F ( x ) ,F ( y )] = 0 , (ix) F ( x ◦ y ) ± ( F ( x ) ◦ F ( y )) = 0 , (x) F ( xy ) ± [ x,y ] ∈ Z ( R ) and (xi) F ( xy ) ± x ◦ y ∈ Z ( R ) for all x,y in some appropriate subset of R , where G : R → R is a multiplicative (generalized)-derivation associated with the map g : R → R . 
Keywords : Semiprime ring, left ideal, derivation, multiplicative derivation, generalized derivation, multiplicative (generalized)-derivation