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Let R be a ring. A generalized semiderivation on R is defined as an additive map \(F:R\rightarrow R\) associated with a semiderivation \(d:R\rightarrow R\) and a map \(g:R\rightarrow R\) which satisfies the conditions \(F(xy) = F(x)y+g(x)d(y) = F(x)g(y)+xd(y)\) and \(F(g(x)) = g(F(x))\) for every \(x, y\in R.\) We show that any generalized semiderivation F on a prime ring R is either a generalized derivation or of the form \(F(b) = \lambda (1-g)(b)\) for every \(b\in R,\) where \(\lambda \in C,\) the extended centroid of R,  1 is the identity map on R and g is an endomorphism.

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On the Structure of Generalized Semiderivations in Prime Rings

  • 会议论文

  • 发表日期：

    2025年

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  • Hafedh Alnoghashi,
  • Nadeemur Rehman

摘要

Let R be a ring. A generalized semiderivation on R is defined as an additive map \(F:R\rightarrow R\) associated with a semiderivation \(d:R\rightarrow R\) and a map \(g:R\rightarrow R\) which satisfies the conditions \(F(xy) = F(x)y+g(x)d(y) = F(x)g(y)+xd(y)\) and \(F(g(x)) = g(F(x))\) for every \(x, y\in R.\) We show that any generalized semiderivation F on a prime ring R is either a generalized derivation or of the form \(F(b) = \lambda (1-g)(b)\) for every \(b\in R,\) where \(\lambda \in C,\) the extended centroid of R,  1 is the identity map on R and g is an endomorphism.