$b$ -Generalized derivations on prime rings

Keywords: Prime ring, $b$-generalized derivation, homomorphism, anti-homomorphism

Abstract

Let $R$ be a prime ring with center $Z(R)$, right Martindale quotient ring $Q$ and extended centroid $C$. By a $b$-generalized derivation we mean an additive mapping $g:R\rightarrow Q$ such that $g(xy) = g(x)y + bxd(y)$ for all $x,y \in R$, where $b \in Q$ and $d: R \rightarrow Q$ is an additive map. In this paper, we extend some well-known results concerning (generalized) derivations on prime rings to $b$-generalized derivations. Further we investigate $b$-generalized derivation acting as a homomorphism or anti-homomorphism in a prime ring.

References

E. Albaş, Generalized derivations on ideals of prime rings, Miskolc Math. Notes, 24, 3 – 9 (2013), https://doi.org/10.18514/mmn.2013.499 DOI: https://doi.org/10.18514/MMN.2013.499

E. Albaş, N. Argaç Generalized derivations of prime rings, Algebra Colloq., 11, № 3, 399 – 410 (2004).

N. Argaç, On prime and semiprime rings with derivations, Algebra Colloq., 13, № 3, 371 – 380 (2006), https://doi.org/10.4134/JKMS.2009.46.5.997 DOI: https://doi.org/10.1142/S1005386706000320

M. Ashraf, A. Asma, A. Shakir, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math., 31, 415 – 421 (2007).

A. Asma, N. Rehman, A. Shakir, On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hung., 101, № 1-2, 79 – 82 (2003), https://doi.org/10.1023/B:AMHU.0000003893.61349.98 DOI: https://doi.org/10.1023/B:AMHU.0000003893.61349.98

K. I. Beidar, W. S. Martindale III, Rings with generalized identities pure and applied mathematics, Dekker, New York (1996).

H. E. Bell, L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hung., 53, 339 – 346 (1989), https://doi.org/10.1007/BF01953371 DOI: https://doi.org/10.1007/BF01953371

M. Bresar, Functional identities of degree two, J. Algebra, 172, 690 – 720 (1995), https://doi.org/10.1006/jabr.1995.1066 DOI: https://doi.org/10.1006/jabr.1995.1066

C. L. Chuang, GPI’s having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103, № 3, 723 – 728 (1988), https://doi.org/10.2307/2046841 DOI: https://doi.org/10.1090/S0002-9939-1988-0947646-4

V. De Filippis, G. Scudo, M. Tammam El-Sayiad, An identity with generalized derivations on Lie ideals, right ideals and Banach algebras, Czechoslovak Math. J., 62, № 137, 453 – 468 (2012), https://doi.org/10.1007/s10587-012-0039-0 DOI: https://doi.org/10.1007/s10587-012-0039-0

B. Dhara, Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings, Beitr. Algebra und Geom., 53, 203 – 209 (2012), https://doi.org/10.1007/s13366-011-0051-9 DOI: https://doi.org/10.1007/s13366-011-0051-9

B. Dhara, S. Kar, K. G. Pradhan, Generalized derivations acting as homomorphism or anti-homomorphism with central values in semiprime rings, Miskolc Math. Notes, 16, № 2, 781 – 791 (2015), https://doi.org/10.18514/MMN.2015.1507 DOI: https://doi.org/10.18514/MMN.2015.1507

B. Hvala, Generalized derivations in prime rings, Comm. Algebra, 26, № 4, 1147 – 1166 (1998), https://doi.org/10.1080/00927879808826190 DOI: https://doi.org/10.1080/00927879808826190

M. T. Koşan, T. K. Lee, $b$-Generalized derivations of semiprime rings having nilpotent values, J. Aust. Math. Soc., 96, 326 – 337 (2014), https://doi.org/10.1017/S1446788713000670 DOI: https://doi.org/10.1017/S1446788713000670

T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20, № 1, 27 – 38 (1992).

E. Posner, Derivations in prime ring, Proc. Amer. Math. Soc., 8, 1093 – 1100 (1957), https://doi.org/10.2307/2032686 DOI: https://doi.org/10.1090/S0002-9939-1957-0095863-0

N. Rehman, M. A. Raza, Generalized derivations as homomorphism and anti-homomorphism on Lie ideals, Arab J. Math. Sci., 22, 22 – 28 (2016), https://doi.org/10.1016/j.ajmsc.2014.09.001 DOI: https://doi.org/10.1016/j.ajmsc.2014.09.001

Y. Wang, H. You, Derivations as homomorphisms or anti-homomorphisms on Lie ideals, Acta Math. Sinica, 23, № 6, 1149 – 1152 (2007), https://doi.org/10.1007/s10114-005-0840-x DOI: https://doi.org/10.1007/s10114-005-0840-x

Published
07.07.2022
How to Cite
PehlivanT., and AlbaşE. “$b$ -Generalized Derivations on Prime Rings”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 6, July 2022, pp. 832 -43, doi:10.37863/umzh.v74i6.5989.
Section
Research articles