Derivations as homomorphisms or anti-homomorphisms in differentially
semiprime rings

M. P. Lukashenko

doi:10.15330/ms.43.1.12-15

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 $d=0$.

1. M. Ashraf, N. Rehman, M. A. Quadri, On $(\sigma,\tau)$–derivations in certain classes of rings, Rad. Mat., 9 (1999), №2, 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), №23, 1105–1110.

3. A. Asma, K. Deepak, Generalized derivations as homomorphisms or as anti–homomorphisms in a prime rings, Hacettepe J. Math., 38 (2009), №1, 17–20.

4. A. Asma, N. Rehman, A. Shakir, On Lie ideals with derivations as homomorphisms and anti– homomorphisms, Acta Math. Hungar. 101 (2003), №1–2, 79–82.

5. H.E. Bell, L.C. Kappe, Rings in which derivations satisfy certain algebraic contradictions, Acta Math. Hungar., 53 (1989), №3–4, 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), №1, 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), №6, 1149–1152.

14. M. Yenigul, N. Argac, On prime and semiprime rings with $\alpha$–derivations, Turkish J. Math., 18 (1994), 280–284.

	Derivations as homomorphisms or anti-homomorphisms in differentially semiprime rings
Author	M. P. Lukashenko bilochka.90@mail.ru Faculty of Mathematics and Informatics, Preсarpathian 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), №2, 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), №23, 1105–1110. 3. A. Asma, K. Deepak, Generalized derivations as homomorphisms or as anti–homomorphisms in a prime rings, Hacettepe J. Math., 38 (2009), №1, 17–20. 4. A. Asma, N. Rehman, A. Shakir, On Lie ideals with derivations as homomorphisms and anti– homomorphisms, Acta Math. Hungar. 101 (2003), №1–2, 79–82. 5. H.E. Bell, L.C. Kappe, Rings in which derivations satisfy certain algebraic contradictions, Acta Math. Hungar., 53 (1989), №3–4, 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), №1, 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), №6, 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
Full text of paper	pdf
Table of content of issue	html

Derivations as homomorphisms or anti-homomorphisms in differentially semiprime rings