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ORTHOGONAL GENERALIZED $(\sigma, \tau)$-DERIVATIONS ON AN IDEAL OF A SEMIPRIME $\Gamma$-RING
Keywords:
semiprime $\Gamma$-ring, generalized $(\sigma, \tau)$-derivation, orthogonal $(\sigma, \tau)$-derivations, orthogonal generalized $(\sigma, \tau)$-derivations.DOI:
https://doi.org/10.17654/0972555524029Abstract
Let $M$ be a $\Gamma$-ring and $\sigma, \tau$ be automorphisms of $M$. An additive mapping $d: M \rightarrow M$ is termed $(\sigma, \tau)$-derivation if $d(u \alpha v)=$ $d(u) \alpha \sigma(v)+\tau(u) \alpha d(v)$ holds for all $u, v \in M$ and $\alpha \in \Gamma$ [10]. A mapping $D: M \rightarrow M$ is deemed a generalized $(\sigma, \tau)$-derivation if there exists $(\sigma, \tau)$-derivation $d: M \rightarrow M$ such that the expression $D(u \alpha v)=D(u) \alpha \sigma(v)+\tau(u) \alpha d(v)$ remains valid for all $u, v \in M$ and $\alpha \in \Gamma$ [10]. This paper builds upon the findings laid out in [9] regarding the orthogonality of $(\sigma, \tau)$-derivations and generalized $(\sigma, \tau)$-derivations within a nonzero ideal of a semiprime ring, extending them to semiprime $\Gamma$-rings.
Received: June 29, 2024
Revised: July 21, 2024
Accepted: August 10, 2024
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