Let R be a semiprime ring, I a non zero ideal of R.
A mapping F:R \longrightarrow R (not necessarily additive) is said to be a multiplicative (generalized)-derivation of R if F\left(xy\right)=F\left(x\right)y+xd\left(y\right) holds for all x, y \in R, where d is any mapping on R. A map H:R\longrightarrow R (not necessarily additive) is called a multiplicative left multiplier if

    \[H\left(xy\right)=H\left(x\right)y, \textnormal{ holds for all } x, y \in R.\]

The main objective of this article is to study the following situations:

\left(i\right)F\left(xoy\right)\pm H\left(xoy\right)=0,

\left(ii\right)F\left(xoy\right)\pm H\left[x,y\right]=0,

\left(iii\right)F\left[x,y\right]\pm \left[x,H\left(y\right)\right]=0,


\left(v\right)F\left(xy\right)\pm \left[x,H\left(y\right)\right]\in Z\left( R\right),

\left(vi\right)F\left(xy\right)\pm\left[H\left(x\right), H\left(y\right)\right]\in Z\left(R\right),

for all x, y in some appropriate subsets of R.

Additional Information


 Malleswari, G. Naga, Shobhalatha, G., Sreenivasulu, S.