Generalized (α,β)-derivations and left Ideals in Prime Rings

Rahaman, Md Hamidur

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creative_2021_30_2_197_202

DOI: https://doi.org/10.37193/CMI.2021.02.10

Issue no: Vol 30/2021 no. 2 Tags: Prime and semiprime rings, generalized $(\alpha, \beta)$-derivations, $(\alpha, automorphisms and ideals

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Let $R$ be a prime ring with center $Z(R)$ , $\lambda$ a nonzero left ideal, $\alpha, \beta$ are automorphisms of $R$ and $R$ admits a generalized $(\alpha,\beta)$ -derivation $F$ associated with a nonzero $(\alpha,\beta)$ -derivation $d$ such that $d(Z(R))\neq (0)$ . In the present paper, we prove that if any one of the following holds:
$(i)$ $F([x,y])-b\alpha(x\circ y)\in Z(R)$
$(ii)$ $F([x,y])+b\alpha(x\circ y)\in Z(R)$
$(iii)$ $F(x \circ y)-b\alpha([x,y])\in Z(R)$
$(iv)$ $F(x \circ y)+b\alpha([x,y])\in Z(R)$
for all $x,y\in \lambda$ and for some $b \in R$ then $R$ is commutative. Also some related results have been obtained.