creative_2021_30_1_61_68_001

Semiprime rings with multiplicative (generalized)- derivations involving left multipliers

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

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creative_2021_30_1_61_68

DOI: https://doi.org/10.37193/CMI.2021.01.07

Issue no: Vol 30/2021 no. 1 Tags: semiprime rings, multiplicative (generalized)-derivations, multiplicative left multipliers, ideal

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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:

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

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

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

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

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

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

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

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Author(s)	Malleswari, G. Naga, Shobhalatha, G., Sreenivasulu, S.