ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS

Authors : Gurninder S SANDHU, Deepak KUMAR

Pages : 87-103

Doi:10.24330/ieja.504124

View : 18 | Download : 8

Publication Date : 2019-01-08

Article Type : Research Paper

Abstract : Let $R$ be a ring, a mapping $F:R\rightarrow R$ together with a mapping $d:R\rightarrow R$ is called a multiplicative insert ignore into journalissuearticles values(generalized);-reverse derivation if $Finsert ignore into journalissuearticles values(xy);=Finsert ignore into journalissuearticles values(y);x+ydinsert ignore into journalissuearticles values(x);$ for all $x,y\in R$. The aim of this note is to investigate the commutativity of prime rings admitting multiplicative insert ignore into journalissuearticles values(generalized);-reverse derivations. Precisely, it is proved that for some nonzero element $a$ in $R$ the conditions: $ainsert ignore into journalissuearticles values(Finsert ignore into journalissuearticles values(xy);\pm xy);=0$, $ainsert ignore into journalissuearticles values(Finsert ignore into journalissuearticles values(x);Finsert ignore into journalissuearticles values(y);\pm xy);=0$, $ainsert ignore into journalissuearticles values(Finsert ignore into journalissuearticles values(xy);\pm Finsert ignore into journalissuearticles values(y);Finsert ignore into journalissuearticles values(x););=0$, $ainsert ignore into journalissuearticles values(Finsert ignore into journalissuearticles values(x);Finsert ignore into journalissuearticles values(y);\pm yx);=0$, $ainsert ignore into journalissuearticles values(Finsert ignore into journalissuearticles values(xy);\pm yx);=0$ are sufficient for the commutativity of $R$. Moreover, we describe the possible forms of generalized reverse derivations of prime rings.
Keywords : Prime ring, multiplicative generalized, reverse derivation, generalized reverse derivation, annihilator conditions