Let \(\mathfrak {R}\) be a ring and \(\mathfrak {L}\) be a square closed Lie ideal of \(\mathfrak {R}\) . A map \(H: \mathfrak {R}\rightarrow \mathfrak {R}\) is called a generalized homoderivation associated with homoderivation h if \(H(\varkappa \vartheta ) = H(\varkappa )h(\vartheta ) + H(\varkappa )\vartheta + \varkappa h(\vartheta )\) is fulfilled for all \(\varkappa ,\vartheta \in \mathfrak {R}\) . The main objective of this paper is to examine the following identities: (1) \(\varkappa H(\vartheta ) \pm \varkappa \vartheta \in \mathfrak {Z}(\mathfrak {R})\) , (2) \(\varkappa H(\vartheta ) \pm \vartheta \varkappa \in \mathfrak {Z}(\mathfrak {R})\) , (3) \(\varkappa H(\vartheta ) \pm [\varkappa ,\vartheta ] \in \mathfrak {Z}(\mathfrak {R})\) , (4) \(H(\vartheta )\varkappa \pm [\varkappa ,\vartheta ] \in \mathfrak {Z}(\mathfrak {R})\) , (5) \([H(\varkappa ),\vartheta ] \pm \varkappa \vartheta \in \mathfrak {Z}(\mathfrak {R})\) , (6) \([H(\varkappa ),\vartheta ] \pm \vartheta \varkappa \in \mathfrak {Z}(\mathfrak {R})\) for all \(\varkappa ,\vartheta \in \mathfrak {L}\) and prove that \(\mathfrak {L}\subseteq \mathfrak {Z}(\mathfrak {R})\) and \(\mathfrak {L}\) is commutative or \(h(\mathfrak {L}) = (0)\) .

On Lie Ideals with Generalized Homoderivations in Prime Rings

会议论文
发表日期：

2025年

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Wasim Ahmed,
Muzibur Rahman Mozumder

摘要

Let \(\mathfrak {R}\) be a ring and \(\mathfrak {L}\) be a square closed Lie ideal of \(\mathfrak {R}\) . A map \(H: \mathfrak {R}\rightarrow \mathfrak {R}\) is called a generalized homoderivation associated with homoderivation h if \(H(\varkappa \vartheta ) = H(\varkappa )h(\vartheta ) + H(\varkappa )\vartheta + \varkappa h(\vartheta )\) is fulfilled for all \(\varkappa ,\vartheta \in \mathfrak {R}\) . The main objective of this paper is to examine the following identities: (1) \(\varkappa H(\vartheta ) \pm \varkappa \vartheta \in \mathfrak {Z}(\mathfrak {R})\) , (2) \(\varkappa H(\vartheta ) \pm \vartheta \varkappa \in \mathfrak {Z}(\mathfrak {R})\) , (3) \(\varkappa H(\vartheta ) \pm [\varkappa ,\vartheta ] \in \mathfrak {Z}(\mathfrak {R})\) , (4) \(H(\vartheta )\varkappa \pm [\varkappa ,\vartheta ] \in \mathfrak {Z}(\mathfrak {R})\) , (5) \([H(\varkappa ),\vartheta ] \pm \varkappa \vartheta \in \mathfrak {Z}(\mathfrak {R})\) , (6) \([H(\varkappa ),\vartheta ] \pm \vartheta \varkappa \in \mathfrak {Z}(\mathfrak {R})\) for all \(\varkappa ,\vartheta \in \mathfrak {L}\) and prove that \(\mathfrak {L}\subseteq \mathfrak {Z}(\mathfrak {R})\) and \(\mathfrak {L}\) is commutative or \(h(\mathfrak {L}) = (0)\) .