Derivations as homomorphisms or antihomomorphisms in differentially semiprime rings 

Author 
bilochka.90@mail.ru
Faculty of Mathematics and Informatics,
Preñarpathian National University of Vasyl Stefanyk

Abstract 
Let $R$ be a ring with an identity, $U$ a nonzero right $d$ideal and $d \in\mathop{\rm Der} R$. We prove that if $R$ is $d$semiprime and $d$ is a homomorphism (respectively an antiendomorphism) of $R$ (respectively acts as a homomorphism on $U$), then $d=0$.
If $R$ is $d$prime and $d$ acts as an antihomomorphism on $U$, then $d=0$.

Keywords 
$d$ideal; homomorphism; differentially semiprime ring

DOI 
doi:10.15330/ms.43.1.1215

Reference 
1. M. Ashraf, N. Rehman, M. A. Quadri, On $(\sigma,\tau)$–derivations in certain classes of rings, Rad. Mat., 9
(1999), ¹2, 187–192.
2. A. Asma, K. Deepak, Derivation which acts as a homomorphism or as an anti–homomorphism in a prime ring, Inter. Math. Forum, 2 (2007), ¹23, 1105–1110. 3. A. Asma, K. Deepak, Generalized derivations as homomorphisms or as anti–homomorphisms in a prime rings, Hacettepe J. Math., 38 (2009), ¹1, 17–20. 4. A. Asma, N. Rehman, A. Shakir, On Lie ideals with derivations as homomorphisms and anti– homomorphisms, Acta Math. Hungar. 101 (2003), ¹1–2, 79–82. 5. H.E. Bell, L.C. Kappe, Rings in which derivations satisfy certain algebraic contradictions, Acta Math. Hungar., 53 (1989), ¹3–4, 339–346. 6. M. Bres.ar, M.A. Chebotar, W.S. Martindale $ 3^{rd}$, Functional identities, Birkh.auser Verlag, Basel Boston Berlin, 2007, 272 p. 7. B. Dhara, Generalized derivations acting as a homomorphism or anti–homomorphism in semiprime rings, Beitr.age Algebra Geom., 53 (2012), ¹1, 203–209. 8. A. Giambruno, I.N. Herstein, Derivations with nilpotent values, Rend. Circ. Mat. Palermo Ser. II, 30 (1981), 199–206. 9. I.N. Herstein, Topics in ring theory, Univ. Chicago Math. Lect. Notes, 1965. 10. N. Rehman, On generalized derivations as homomorphisms and anti–homomorphisms, Glasnik Mat., 39(59) (2004), 27–30. 11. N. Rehman, M.A. Raza, Generalized derivations as homomorphisms and anti–homomorphisms on Lie ideals, Arab J. Math. Sci., (to appear) http://dx.doi.org/10.1016/j.amsc.2014.09.001. 12. G. Scudo, Generalized derivations acting as homomorphisms on polynomials in prime rings, Southeast Asian Bull. Math., 38 (2014), 563–572. 13. Y. Wang, H. You, Derivations as homomorphisms or anti–homomorphisms on Lie ideals, Acta Math. Sin., 23 (2007), ¹6, 1149–1152. 14. M. Yenigul, N. Argac, On prime and semiprime rings with $\alpha$–derivations, Turkish J. Math., 18 (1994), 280–284. 
Pages 
1215

Volume 
43

Issue 
1

Year 
2015

Journal 
Matematychni Studii

Full text of paper  
Table of content of issue 