## Rocky Mountain Journal of Mathematics

### Two sided $\alpha$-derivations in 3-prime near-rings

#### Abstract

The purpose of this paper is to investigate two sided $\alpha$-derivations satisfying certain differential identities on 3-prime near-rings. Some well-known results characterizing commutativity of 3-prime near-rings by derivations (semi-derivations) have been generalized. Furthermore, examples proving the necessity of the 3-primeness hypothesis are given.

#### Article information

Source
Rocky Mountain J. Math., Volume 46, Number 4 (2016), 1379-1393.

Dates
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1476884588

Digital Object Identifier
doi:10.1216/RMJ-2016-46-4-1379

Mathematical Reviews number (MathSciNet)
MR3563187

Zentralblatt MATH identifier
1353.16018

#### Citation

Samman, M.; Oukhtite, L.; Raji, A.; Boua, A. Two sided $\alpha$-derivations in 3-prime near-rings. Rocky Mountain J. Math. 46 (2016), no. 4, 1379--1393. doi:10.1216/RMJ-2016-46-4-1379. https://projecteuclid.org/euclid.rmjm/1476884588

#### References

• N. Argaç, On near-rings with two sided $\alpha$-derivations, Turkish J. Math. 28 (2004), 195–204.
• M. Ashraf and A. Shakir, On $(\sigma, \tau)$-derivations of prime near-rings II, Sarajevo J. Math. 4 (2008), 23–30.
• H.E. Bell, On derivations in near-rings II, Kluwer Academic Publishers, Netherlands, 1997.
• H.E. Bell and N. Argaç, Derivations, products of derivations, and commutativity in near-rings, Alg. Colloq. 8 (2001), 399–407.
• H.E. Bell, A. Boua and L. Oukhtite, Semigroup ideals and commutativity in $3$-prime near rings, Comm. Alg. 43 (2015), 1757–1770.
• H.E. Bell and G. Mason, On derivations in near-rings, North-Holland Math. Stud. 137 (1987), 31–35.
• ––––, On derivations in near-rings and rings, Math. J. Okayama Univ. 34 (1992), 135–144.
• J. Bergen, Derivations in prime rings, Canad. Math. Bull. 26 (1983), 267–270.
• A. Boua and L. Oukhtite, Derivations on prime near-rings, Int. J. Open Prob. Comp. Sci. Math. 4 (2011), 162–167.
• ––––, Semiderivations satisfying certain algebraic identities on prime near-rings, Asian-Europ. J. Math. 6 (2013), 1350043 (8 pages).
• A. Boua, L. Oukhtite and H.E. Bell, Differential identities on semigroup ideals of right near-rings, Asian-Europ. J. Math. 6 (2013), DOI: 10.1142/S1793557113500502.
• M.N. Daif and H.E. Bell, Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci. 15 (1992), 205–206.
• M. Hongan, On near-rings with derivations, Math. J. Okayama Univ. 32 (1990), 89–92.
• X.K. Wang, Derivations in prime near-rings, Proc. Amer. Math. Soc. 121 (1994), 361–366.