Posner's first theorem for prime modules

Document Type : Original Article

Authors

Hoger Ghahramani

Mohammad Nader Ghosseiri

Tahereh Rezaei

Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran

10.22034/kjm.2024.398628.2879

Abstract

‎Let $R$ be a ring‎, ‎$M$ be a left $R$-module‎, ‎$\tau:M\rightarrow M$ and $\delta:R\rightarrow R$ be additive maps‎. ‎We say that $\tau$ is a generalized derivation relative to $\delta$ if $\tau(am)=a\tau(m)+\delta(a)m$ for all $a\in R$ and $m\in M$‎. ‎In this paper we provide a generalization of Posner's first theorem to generalized derivations on ${2}$-torsion free prime modules‎. ‎We also obtain a result of this generalization in connection with derivations acting on left ideals of prime rings‎. ‎Moreover‎, ‎we extend some previous results related to Posner's first theorem‎. ‎Further‎, ‎as an application of our main result‎, ‎we examine Posner's first theorem for a certain class of derivations on trivial extension rings under some suitable conditions‎.

Keywords

Generalized derivations‎

‎prime module‎

‎Posner's first theorem

Subjects

• 16 Associative rings and algebras

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