# Institute of Mathematics

## Generalized derivations acting on multilinear polynomials in prime rings

#### Basudeb Dhara

###### Received July 6, 2016.   First published December 5, 2017.

Abstract:  Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$, let $F$, $G$ and $H$ be three generalized derivations of $R$, $I$ an ideal of $R$ and $f(x_1,\ldots,x_n)$ a multilinear polynomial over $C$ which is not central valued on $R$. If $F(f(r))G(f(r))=H(f(r)^2)$ for all $r=(r_1,\ldots,r_n) \in I^n$, then one of the following conditions holds: \item{(1)} there exist $a\in C$ and $b\in U$ such that $F(x)=ax$, $G(x)=xb$ and $H(x)=xab$ for all $x\in R$; \item{(2)} there exist $a, b\in U$ such that $F(x)=xa$, $G(x)=bx$ and $H(x)=abx$ for all $x\in R$, with $ab\in C$; \item{(3)} there exist $b\in C$ and $a\in U$ such that $F(x)=ax$, $G(x)=bx$ and $H(x)=abx$ for all $x\in R$; \item{(4)} $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and one of the following conditions holds: \itemitem{(a)} there exist $a,b,p,p'\in U$ such that $F(x)=ax$, $G(x)=xb$ and $H(x)=px+xp'$ for all $x\in R$, with $ab=p+p'$; \itemitem{(b)} there exist $a,b,p,p'\in U$ such that $F(x)=xa$, $G(x)=bx$ and $H(x)=px+xp'$ for all $x\in R$, with $p+p'=ab\in C$.
Keywords:  prime ring; derivation; generalized derivation; extended centroid; Utumi quotient ring
Classification MSC:  16W25, 16N60
DOI:  10.21136/CMJ.2017.0352-16

References:
[1] E. Albaş: Generalized derivations on ideals of prime rings. Miskolc Math. Notes 14 (2013), 3-9. MR 3070683 | Zbl 1289.16082
[2] S. Ali, S. Huang: On generalized Jordan $(\alpha,\beta)$-derivations that act as homomorphisms or anti-homomorphisms. J. Algebra Comput. Appl. (electronic only) 1 (2011), 13-19. MR 2862508 | Zbl 1291.16038
[3] N. Argaç, V. De Filippis: Actions of generalized derivations on multilinear polynomials in prime rings. Algebra Colloq. 18, Spec. Iss. 1 (2011), 955-964. DOI 10.1142/S1005386711000836 | MR 2860377 | Zbl 1297.16037
[4] A. Asma, N. Rehman, A. Shakir: On Lie ideals with derivations as homomorphisms and anti-homomorphisms. Acta Math. Hungar 101 (2003), 79-82. DOI 10.1023/B:AMHU.0000003893.61349.98 | MR 2011464 | Zbl 1053.16025
[5] H. E. Bell, L. C. Kappe: Rings in which derivations satisfy certain algebraic conditions. Acta Math. Hung. 53 (1989), 339-346. DOI 10.1007/BF01953371 \goodbreak | MR 1014917 | Zbl 0705.16021
[6] J. Bergen, I. N. Herstein, J. W. Keer: Lie ideals and derivations of prime rings. J. Algebra 71 (1981), 259-267. DOI 10.1016/0021-8693(81)90120-4 | MR 0627439 | Zbl 0463.16023
[7] L. Carini, V. De Filippis, G. Scudo: Identities with product of generalized derivations of prime rings. Algebra Colloq. 20 (2013), 711-720. DOI 10.1142/S1005386713000680 | MR 3116800 | Zbl 1285.16036
[8] C.-L. Chuang: The additive subgroup generated by a polynomial. Isr. J. Math. 59 (1987), 98-106. DOI 10.1007/BF02779669 | MR 0923664 | Zbl 0631.16015
[9] C.-L. Chuang: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103 (1988), 723-728. DOI 10.2307/2046841 | MR 0947646 | Zbl 0656.16006
[10] V. De Filippis: Generalized derivations as Jordan homomorphisms on Lie ideals and right ideals. Acta Math. Sin., Engl. Ser. 25 (2009), 1965-1974. DOI 10.1007/s10114-009-7343-0 | MR 2578635 | Zbl 1192.16042
[11] V. De Filippis, O. M. Di Vincenzo: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40 (2012), 1918-1932. DOI 10.1080/00927872.2011.553859 | MR 2945689 | Zbl 1258.16043
[12] V. De Filippis, G. Scudo: Generalized derivations which extend the concept of Jordan homomorphism. Publ. Math. 86 (2015), 187-212. DOI 10.5486/PMD.2015.7070 | MR 3300586 | Zbl 1341.16040
[13] B. Dhara: Derivations with Engel conditions on multilinear polynomials in prime rings. Demonstr. Math. 42 (2009), 467-478. MR 2554943 | Zbl 1188.16037
[14] B. Dhara: Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings. Beitr. Algebra Geom. 53 (2012), 203-209. DOI 10.1007/s13366-011-0051-9 | MR 2890375 | Zbl 1242.16039
[15] B. Dhara, S. Huang, A. Pattanayak: Generalized derivations and multilinear polynomials in prime rings. Bull. Malays. Math. Sci. Soc. 36 (2013), 1071-1081. MR 3108796 | Zbl 1281.16046
[16] B. Dhara, N. U. Rehman, M. A. Raza: Lie ideals and action of generalized derivations in rings. Miskolc Math. Notes 16 (2015), 769-779. DOI 10.18514/MMN.2015.1343 | MR 3454141 | Zbl 1349.16068
[17] B. Dhara, S. Sahebi, V. Rehmani: Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals and right ideals. Math. Slovaca 65 (2015), 963-974. DOI 10.1515/ms-2015-0065 | MR 3433047 | Zbl 06534094
[18] T. S. Erickson, W. S. Martindale III, J. M. Osborn: Prime nonassociative algebras. Pac. J. Math. 60 (1975), 49-63. DOI 10.2140/pjm.1975.60.49 | MR 0382379 | Zbl 0355.17005
[19] I. Gusić: A note on generalized derivations of prime rings. Glas. Mat., III. Ser. 40 (2005), 47-49. DOI 10.3336/gm.40.1.05 | MR 2195859 | Zbl 1072.16031
[20] N. Jacobson: Structure of Rings. American Mathematical Society Colloquium Publications 37, Revised edition American Mathematical Society, Providence (1956). DOI 10.1090/coll/037 | MR 0222106 | Zbl 0073.02002
[21] V. K. Kharchenko: Differential identities of prime rings. Algebra Logic 17 (1978), 155-168. (In English. Russian original.); translation from Algebra Logika 17 (1978), 220-238. DOI 10.1007/BF01670115 | Zbl 0423.16011
[22] C. Lanski: Differential identities, Lie ideals, and Posner's theorems. Pac. J. Math. 134 (1988), 275-297. DOI 10.2140/pjm.1988.134.275 | MR 0961236 | Zbl 0614.16028
[23] C. Lanski: An Engel condition with derivation. Proc. Am. Math. Soc. 118 (1993), 731-734. DOI 10.2307/2160113 | MR 1132851 | Zbl 0821.16037
[24] T.-K. Lee: Semiprime rings with differential identities. Bull. Inst. Math., Acad. Sin. 20 (1992), 27-38. MR 1166215 | Zbl 0769.16017
[25] T.-K. Lee: Generalized derivations of left faithful rings. Commun. Algebra 27 (1999), 4057-4073. DOI 10.1080/00927879908826682 | MR 1700189 | Zbl 0946.16026
[26] P.-H. Lee, T.-K. Lee: Derivations with Engel conditions on multilinear polynomials. Proc. Am. Math. Soc. 124 (1996), 2625-2629. DOI 10.1090/S0002-9939-96-03351-5 | MR 1327023 | Zbl 0859.16031
[27] U. Leron: Nil and power central polynomials in rings. Trans. Am. Math. Soc. 202 (1975), 97-103. DOI 10.2307/1997300 | MR 0354764 | Zbl 0297.16010
[28] W. S. Martindale III: Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969), 576-584. DOI 10.1016/0021-8693(69)90029-5 | MR 0238897 | Zbl 0175.03102
[29] E. C. Posner: Derivations in prime rings. Proc. Am. Math. Soc. 8 (1957), 1093-1100. DOI 10.2307/2032686 | MR 0095863 | Zbl 0082.03003
[30] N. U. Rehman: On generalized derivations as homomorphisms and anti-homomorphisms. Glas. Mat., III. Ser. 39 (2004), 27-30. DOI 10.3336/gm.39.1.03 | MR 2055383 | Zbl 1047.16019
[31] Y. Wang, H. You: Derivations as homomorphisms or anti-homomorphisms on Lie ideals. Acta Math. Sin., Engl. Ser. 23 (2007), 1149-1152. DOI 10.1007/s10114-005-0840-x | MR 2319944 | Zbl 1124.16031

Affiliations:   Basudeb Dhara, Department of Mathematics, Belda College, Belda, Paschim Medinipur, 721424, West Bengal, India, e-mail: basu_dhara@yahoo.com

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