Derivations as homomorphisms or anti-homomorphisms in differentially semiprime rings

Author
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 anti-endomorphism) of $R$ (respectively acts as a homomorphism on $U$), then $d=0$. If $R$ is $d$-prime and $d$ acts as an anti-homomorphism on $U$, then $d=0$.
Keywords
$d$-ideal; homomorphism; differentially semiprime ring
DOI
doi:10.15330/ms.43.1.12-15
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
12-15
Volume
43
Issue
1
Year
2015
Journal
Matematychni Studii
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