Jordan homoderivation behavior of generalized derivations in prime rings

Nripendu Bera; Basudeb Dhara

doi:10.3842/umzh.v75i9.7241

Nripendu Bera Department of Mathematics, Jadavpur University, Kolkata, India
Basudeb Dhara Department of Mathematics, Belda College, Belda, Paschim Medinipur, India

DOI: https://doi.org/10.3842/umzh.v75i9.7241

Keywords: Generalized derivation, Homoderivation, Utumi quotient ring, Extended centroid, Multilinear polynomial, Prime ring

Abstract

UDC 512.5

Suppose that $R$ is a prime ring with ${\rm char}(R)\neq 2$ and $f(\xi_1,\ldots,\xi_n)$ is a noncentral multilinear polynomial over $C( = Z(U)),$ where $U$ is the Utumi quotient ring of $R.$ An additive mapping $h\colon R\rightarrow R$ is called homoderivation if $h(ab) = h(a)h(b)+h(a)b+ah(b)$ for all $a,b\in R.$ We investigate the behavior of three generalized derivations $F,$ $G,$ and $H$ of $R$ satisfying the condition $$F(\xi^2) = G(\xi)^2+H(\xi)\xi+\xi H(\xi)$$ for all $\xi \in f(R) = \{f(\xi_1,\ldots,\xi_n) \mid \xi_1,\ldots,\xi_n\in R\}.$

References

E. Albas, Generalized derivations on ideals of prime rings, Miskolc Math. Notes, 14, № 1, 3–9 (2013). DOI: https://doi.org/10.18514/MMN.2013.499

E. F. Alharfie, N. M. Muthana, On homoderivations and commutativity of rings, Bull. Inst. Math. Virtual Inst., 9, 301–304 (2019).

E. F. Alharfie, N. M. Muthana, The commutativity of prime rings with homoderivations, Int. J. Adv. and Appl. Sci., 5, № 5, 79–81 (2018). DOI: https://doi.org/10.21833/ijaas.2018.05.010

A. Al-Kenani, A. Melaibari, N. Muthana, Homoderivations and commutativity of $ast$-prime rings, East-West J. Math., 17, № 2, 117–126 (2015).

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

K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with generalized identities, Pure and Appl. Math., 196, Marcel Dekker, New York (1996).

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

V. De Filippis, O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra, 40, № 6, 1918–1932 (2012). DOI: https://doi.org/10.1080/00927872.2011.553859

V. De Filippis, Generalized derivations as Jordan homomorphisms on Lie ideals and right ideals, Acta Math. Sin. (Engl. Ser.), 25, № 2, 1965–1974 (2009). DOI: https://doi.org/10.1007/s10114-009-7343-0

B. Dhara, Generalized derivations acting on multilinear polynomials in prime rings, Czech. Math. J., 68, № 1, 95–119 (2018). DOI: https://doi.org/10.21136/CMJ.2017.0352-16

B. Dhara, S. Sahebi, V. Rehmani, Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals and right ideals, Math. Slovaca, 65, № 5, 963–974 (2015). DOI: https://doi.org/10.1515/ms-2015-0065

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

M. M. El Sofy, Rings with some kinds of mappings, M. Sci. Thesis, Cairo Univ., Branch of Fayoum, Cairo, Egypt (2000).

C. Faith, Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hung., 14, 369–371 (1963). DOI: https://doi.org/10.1007/BF01895723

I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8, 1104–1110 (1957). DOI: https://doi.org/10.1090/S0002-9939-1957-0095864-2

N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Publ., 37 (1964).

V. K. Kharchenko, Differential identity of prime rings, Algebra and Logic, 17, 155–168 (1978). DOI: https://doi.org/10.1007/BF01670115

T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra, 27, № 8, 4057–4073 (1999). DOI: https://doi.org/10.1080/00927879908826682

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

W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12, 576–584 (1969). DOI: https://doi.org/10.1016/0021-8693(69)90029-5

A. Melaibari, N. Muthana, A. Al-Kenani, Centrally-extended homoderivations on rings, Gulf J. Math., 4, № 2, 62–70 (2016). DOI: https://doi.org/10.56947/gjom.v4i2.67

T. L. Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq., 3, 369–478 (1996).

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