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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2017 / Volume: 46 Issue: 6

Herstein’s theorem for generalized derivations in rings with involution

Shakir Ali
Abdul Nadim Khan
Nadeem Ahmad Dar
Sayfa
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Yayın Tarihi
2017-12-01
DOI
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In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Let $R$ be an associative ring.
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Temel Bulgular & Veri:In [7], Herstein proved the following result: If $R$ is a prime ring of $char(R)\neq 2$ admitting a nonzero derivation $d$ such that $[d(x),d(y)]=0$ for all $x,y\in R$, then $R$ is commutative.
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Sonuç & Katkı:In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Prime ring · involution · derivation

Abstract

Let $R$ be an associative ring. An additive map $F:R\toR$ is called a generalized derivation if there exists a derivation $d$ of $R$ such that $F(xy)=F(x)y+xd(y)$ for all $x,y\in R$. In [7], Herstein proved the following result: If $R$ is a prime ring of $char(R)\neq 2$ admitting a nonzero derivation $d$ such that $[d(x),d(y)]=0$ for all $x,y\in R$, then $R$ is commutative. In the present paper, we shall study the above mentioned result for generalized derivations in rings with involution.

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Bu Makaleye Atıf Yap

Ali, S., Khan, A. N., Dar, N. A. (2017). Herstein’s theorem for generalized derivations in rings with involution. Hacettepe Journal of Mathematics and Statistics, Volume: 46 Issue: 6. https://ajindex.com/articles/1402983

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Kaynakça (9)

  1. Ali, S. and Dar, N. A. On -centralizing mappings in rings with involution, Georgian Math. J. 21(1), 25–28, 2014.
  2. Bell, H. E. and Rehman, N. Generalized derivations with commutattivity and anticommutativity conditions, Math. J. Okayama Univ. 49, 139–147, 2007.
  3. Brešar, M. On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33, 89–93, 1991.
  4. Daif, M. N. Commutativity results for semiprime rings with derivation, Int. J. Math. and Math. Sci. 21(3), 471–474, 1998.
  5. Dar, N. A. and Ali, S. On -commuting mappings and derivaitons in rings with involution, Turkish J. Math. 40, 884–894, 2016.
  6. Herstein, I. N. Rings with involution (The university of Chicago Press, 1976).
  7. Herstein, I. N. A note on derivations, Canad. Math. Bull. 21, 369–370, 1978.
  8. Hvala, B. Generalized derivations in rings, Comm. Algebra 26(4), 1147–1166, 1998.
  9. Rehman, N. and De Filippis, V. Commutativity and skew-commutativity conditions with generalized derivations, Algebra Colloq. 17, 841-850, 2010.