$(\theta,\phi)$-derivations as homomorphisms or as anti-homomorphisms on a near ring

Main Article Content

Asma Ali
Howard E. Bell
Rekha Rani


Let $N$ be a near ring. An additive mapping $d:N\longrightarrow N$ is said to be a $(\theta,\phi)$-derivation on $N$ if there exist mappings $\theta,\phi:N\longrightarrow N$ such that$d(xy)=\theta(x)d(y)+d(x)\phi(y)$ holds for all $x,y \in N$. In the context of 3-prime and 3-semiprime nearrings, we show that for suitably-restricted $\theta$ and $\phi$, there exist no nonzero $(\theta,\phi)$-derivations which act as a homomorphism or an anti-homomorphism on $N$ or a nonzero semigroup ideal of $N$.

Article Details

How to Cite
Ali, A., Bell, H. E., & Rani, R. (2012). $(\theta,\phi)$-derivations as homomorphisms or as anti-homomorphisms on a near ring. Tamkang Journal of Mathematics, 43(3), 385–390. https://doi.org/10.5556/j.tkjm.43.2012.849
Author Biographies

Asma Ali, Aligarh Muslim University, Aligarh-202002, (UP) India

Department of mathematics, Associate professor

Howard E. Bell, Department of mathematics, Brock University, St. Catharines, Ontario L2S 3A1, Canada

Department of mathematics, Professor

Rekha Rani, N. R. E. C. College, Khurja-203131, (UP) India

Department of mathematics, Assistant Professor


M. Ashraf, Ali Asma and Ali Shakir, ($sigma,tau$)-derivations on prime near rings, Arch. Math., 40(2004), 281--286.

A. Ali, N. Rehman and A. Shakir, On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar., 101(2003), 79--82.

H. E. Bell, On derivations in near rings II, Kluwer Academic Publ. Math. Appl. Dordr., 426(1997), 191--197.

H. E. Bell and L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta. Math. Hungar., 53(1989), 339--346.

X. K. , Wang, Derivations in prme near rings, proc. Amer. Math. Soc., 121(1994), 361--366.