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

Authors : Asma ALİ, Basudeb DHARA, Shahoor KHAN

Pages : 1293-1306

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Publication Date : 2015-12-01

Article Type : Research Paper

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