Some identities involving multiplicative semiderivations on ideals

Authors : Oznur GOLBASİ, Zeliha BEDİR

Pages : 963-969

Doi:10.15672/hujms.650600

View : 17 | Download : 7

Publication Date : 2021-08-06

Article Type : Research Paper

Abstract :Let $R$ be a prime ring and $I$ be a nonzero ideal of $R.$ A mapping $d:R\rightarrow R$ is called a multiplicative semiderivation if there exists a function $g:R\rightarrow R$ such that insert ignore into journalissuearticles values(i); $dinsert ignore into journalissuearticles values(xy);=dinsert ignore into journalissuearticles values(x);ginsert ignore into journalissuearticles values(y);+xdinsert ignore into journalissuearticles values(y);=dinsert ignore into journalissuearticles values(x);y+ginsert ignore into journalissuearticles values(x);dinsert ignore into journalissuearticles values(y);$ and insert ignore into journalissuearticles values(ii); $dinsert ignore into journalissuearticles values(ginsert ignore into journalissuearticles values(x););=ginsert ignore into journalissuearticles values(dinsert ignore into journalissuearticles values(x););$ hold for all $x,y\in R.$ In the present paper, we shall prove that $[x,dinsert ignore into journalissuearticles values(x);]=0,$ for all $x\in I$ if any of the followings holds: i); $dinsert ignore into journalissuearticles values(xy);\pm xy\in Z,$ ii); $dinsert ignore into journalissuearticles values(xy);\pm yx\in Z,$ iii); $dinsert ignore into journalissuearticles values(x);dinsert ignore into journalissuearticles values(y);\pm xy\in Z,$ iv); $dinsert ignore into journalissuearticles values(xy);\pm dinsert ignore into journalissuearticles values(x);dinsert ignore into journalissuearticles values(y);\in Z,$ viii); $dinsert ignore into journalissuearticles values(xy);\pm dinsert ignore into journalissuearticles values(y);dinsert ignore into journalissuearticles values(x);\in Z,$ for all $x,y\in I.$ Also, we show that $R$ must be commutative if $dinsert ignore into journalissuearticles values(I);\subseteq Z.$
Keywords : multiplicative semiderivation, prime rings, semiderivation