## Rocky Mountain Journal of Mathematics

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

Tsiu-Kwen Lee

#### Abstract

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

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

Dates
First available in Project Euclid: 18 April 2017

https://projecteuclid.org/euclid.rmjm/1492502548

Digital Object Identifier
doi:10.1216/RMJ-2017-47-2-511

Mathematical Reviews number (MathSciNet)
MR3635372

Zentralblatt MATH identifier
1371.16020

#### Citation

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. https://projecteuclid.org/euclid.rmjm/1492502548

#### References

• 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.