Rocky Mountain Journal of Mathematics

Jordan [! \large !]$\sigma $-derivations of prime rings

Tsiu-Kwen Lee

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $R$ be a noncommutative prime ring with extended centroid~$C$ and with $Q_{mr}(R)$ its maximal right ring of quotients. From the viewpoint of functional identities, we give a complete characterization of Jordan $\sigma $-derivations of $R$ with $\sigma $ an epimorphism. Precisely, given such a Jordan $\sigma $-derivation $\de \colon R\to Q_{mr}(R)$, it is proved that either $\delta $ is a $\sigma $-derivation or a derivation $d\colon R\to Q_{mr}(R)$ and a unit $u\in Q_{mr}(R)$ exist such that $\delta (x)=ud(x)+\mu (x)u$ for all $x\in R$, where $\mu \colon R\to C$ is an additive map satisfying $\mu (x^2)=0$ for all $x\in R$. In addition, if $\sigma $ is an X-outer automorphism, then $\delta $ is always a $\sigma $-derivation.

Article information

Rocky Mountain J. Math., Volume 47, Number 2 (2017), 511-525.

First available in Project Euclid: 18 April 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16N60: Prime and semiprime rings [See also 16D60, 16U10] 16R60: Functional identities 16W25: Derivations, actions of Lie algebras

Prime ring (Jordan) $\sigma $-derivation PI GPI functional identity


Lee, Tsiu-Kwen. Jordan [! \large !]$\sigma $-derivations of prime rings. Rocky Mountain J. Math. 47 (2017), no. 2, 511--525. doi:10.1216/RMJ-2017-47-2-511.

Export citation


  • K.I. Beidar, M. Brešar and M.A. Chebotar, Generalized functional identities with $($anti-$)$automorphisms and derivations on prime rings, I, J. Algebra 215 (1999), 644–665.
  • K.I. Beidar and W.S. Martindale, III, On functional identities in prime rings with involution, J. Algebra 203 (1998), 491–532.
  • K.I. Beidar, W.S. Martindale, III, and A.A. Mikhalev, Rings with generalized identities, in Monographs and textbooks in pure and applied mathematics, 196, Marcel Dekker, Inc., New York, 1996.
  • M. Brešar, M.A. Chebotar and W.S. Martindale, III, Functional identities, in Frontiers in mathematics, Birkhauser Verlag, Basel, 2007.
  • V. De Filippis, A. Mamouni and L. Oukhtite, Generalized Jordan semiderivations in prime rings, Canad. Math. Bull. 58 (2015), 263–270.
  • I.N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.
  • V.K. Kharchenko, Generalized identities with automorphisms, Alg. Logik. 14 (1975), 215–237.
  • T.-K. Lee, Functional identities and Jordan $\sigma$-derivations, Linear Multilin. Alg. 64 (2016), 221–234.
  • T.-K. Lee and J.-H. Lin, Jordan derivations of prime rings with characteristic two, Lin. Alg. Appl. 462 (2014), 1–15.
  • T.-K. Lee and K.-S. Liu, The Skolem-Noether theorem for semiprime rings satisfying a strict identity, Comm. Alg. 35 (2007), 1949–1955.
  • W.S. Martindale, III, Prime rings satisfying a generalized polynomial identity, J. Alg. 12 (1969), 576–584.
  • S. Montgomery, Fixed rings of finite automorphism groups of associative rings, Lect. Notes Math. 818, Springer, Berlin, 1980.
  • L. Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219–223.