Generalized derivations acting on multilinear polynomials as a Jordan homomorphisms
Анотація
УДК 512.5
Узагальненi похiднi, що дiють на мультилiнiйних полiномахяк жордановi гомоморфiзми
Нехай $R$ — просте кільце з характеристикою, що не дорівнює $2,$ $U$ — фактор-кільце Утумі для $R,$ а $C$ — продовжений центроїд для $R.$ Крім того, припустимо, що $G$ та $H$ — дві узагальнені похідні на $R,$ а $f(x_1,\ldots,x_n)$ — нецентральний мультилінійний поліном над $C.$ Якщо $G(H(u^2))=(H(u))^2$ для всіх $u=f(r_1,\ldots,r_n),$ $r_1,\ldots,r_n \in R,$ то справджується одне з таких тверджень:
1) $H=0;$
2) існує таке $\lambda\in C,$ що $G(x)=H(x)=\lambda x$ для всіх $x\in R;$
3) існують такі $\lambda\in C$ та $a\in U,$ що $H(x)=\lambda x,$ $G(x)=[a, x]+\lambda x$ для всіх $x\in R$ і $f(x_1,\ldots,x_n)^2$ є центральнозначним на $R.$
Посилання
V. De Filippis, Generalized derivations as Jordan homomorphisms on lie ideals and right ideals, Acta Math. Sin. (Engl. Ser.), 25, № 12, 1965 – 1974 (2009), https://doi.org/10.1007/s10114-009-7343-0 DOI: https://doi.org/10.1007/s10114-009-7343-0
V. De Filippis, Generalized skew derivations as Jordan homomorphisms on multilinear polynomials, J. Korean Math. Soc., 52, № 1, 191 – 207 (2009), https://doi.org/10.4134/JKMS.2015.52.1.191 DOI: https://doi.org/10.4134/JKMS.2015.52.1.191
V. De Filippis, G. Scudo, Generalized derivations which extend the concept of Jordan homomorphism, Publ. Math. Debrecen, 86, № 1-2, 187 – 212 (2015), https://doi.org/10.5486/PMD.2015.7070 DOI: https://doi.org/10.5486/PMD.2015.7070
V. De Filippis, B. Dhara, Generalized skew-derivations and generalization of homomorphism maps in prime rings, Comm. Algebra, 47, № 8, 3154 – 3169 (2019), https://doi.org/10.1080/00927872.2018.1552285 DOI: https://doi.org/10.1080/00927872.2018.1552285
B. Dhara, Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings, Beitr. Algebra and 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, Generalized derivations acting on multilinear polynomials in prime rings, Czechoslovak Math. J., 68, № 1, 95 – 119 (2018), https://doi.org/10.1007/s13366-011-0051-9 DOI: https://doi.org/10.21136/CMJ.2017.0352-16
K. I. Beidar, W. S. Martindale III, V. Mikhalev, Rings with generalized identities, Marcel Dekker, New York (1996).
H. E. Bell, L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar., 53, 339 – 346 (1989). DOI: https://doi.org/10.1007/BF01953371
L. Carini, V. De Filippis, G. Scudo, Identities with product of generalized skew derivations on multilinear polynomials, Comm. Algebra, 44, № 7, 3122 – 3138 (2016), https://doi.org/10.1080/00927872.2015.1027354 DOI: https://doi.org/10.1080/00927872.2015.1027354
C. L. Chuang, GPIs 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
T. S. Erickson, W. S. Martindale III, J. M. Osborn, Prime nonassociative algebras, Pacif. J. Math., 60, 49 – 63 (1975). DOI: https://doi.org/10.2140/pjm.1975.60.49
C. Faith, Y. Utumi, On a new proof of Litoff ’s theorem, Acta Math. Acad. Sci. Hung., 14, 369 – 371 (1963), https://doi.org/10.1007/BF01895723 DOI: https://doi.org/10.1007/BF01895723
I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc., 81, 331 – 341 (1956), https://doi.org/10.2307/1992920 DOI: https://doi.org/10.1090/S0002-9947-1956-0076751-6
N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Publ., 37, Amer. Math. Soc., Providence, RI (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), https://doi.org/10.1080/00927879908826682 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).
U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc., 202, 97 – 103 (1975), https://doi.org/10.2307/1997300 DOI: https://doi.org/10.1090/S0002-9947-1975-0354764-6
W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12, 576 – 584 (1969), https://doi.org/10.1016/0021-8693(69)90029-5 DOI: https://doi.org/10.1016/0021-8693(69)90029-5
M. F. Smiley, Jordan homomorphisms onto prime rings, Trans. Amer. Math. Soc., 84, 426 – 429 (1957), https://doi.org/10.2307/1992823 DOI: https://doi.org/10.1090/S0002-9947-1957-0083484-X
E. Albas, N. Argac, Generalized derivations of prime rings, Algebra Colloq., 11, 399 – 410 (2004).
A. Ali, N. Rehman, S. Ali, On lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar., 101, 79 – 82 (2003), https://doi.org/10.1023/B:AMHU.0000003893.61349.98 DOI: https://doi.org/10.1023/B:AMHU.0000003893.61349.98
S. K. Tiwari, Generalized derivations with multilinear polynomials in prime rings, Comm. Algebra, 46, № 12, 5356 – 5372 (2018), https://doi.org/10.1080/00927872.2018.1468899 DOI: https://doi.org/10.1080/00927872.2018.1468899
S. K. Tiwari, R. K. Sharma, B. Dhara, Identities related to generalized derivation on ideal in prime rings, Beitr. Algebra and Geom., 57, № 4, 809 – 821 (2016), https://doi.org/10.1007/s13366-015-0262-6 DOI: https://doi.org/10.1007/s13366-015-0262-6
S. K. Tiwari, R. K. Sharma, B. Dhara, Multiplicative (generalized)-derivation in semiprime rings, Beitr. Algebra and Geom., 58, № 1, 211 – 225 (2017), https://doi.org/10.1007/s13366-015-0279-x DOI: https://doi.org/10.1007/s13366-015-0279-x
Авторські права (c) 2022 S. K. Tiwari, B Prajapati
![Creative Commons License](http://i.creativecommons.org/l/by/4.0/88x31.png)
Для цієї роботи діють умови ліцензії Creative Commons Attribution 4.0 International License.