Previous |  Up |  Next

# Article

Full entry | PDF   (0.2 MB)
Keywords:
skew derivation; generalized polynomial identity (GPI); prime ring; ideal
Summary:
Let $R$ be a prime ring with center $Z$ and $I$ be a nonzero ideal of $R$. In this manuscript, we investigate the action of skew derivation $(\delta,\varphi)$ of $R$ which acts as a homomorphism or an anti-homomorphism on $I$. Moreover, we provide an example for semiprime case.
References:
[1] Asma A., Rehman N., Ali S.: 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
[2] Beidar K.I., Martindale W.S.III, Mikhalev A.V.: Rings with Generalized Identities. Pure and Applied Mathematics, 196, Marcel Dekker, New York, 1996. MR 1368853 | Zbl 0847.16001
[3] Bell H.E., Kappe L.C.: Rings in which derivations satisfy certain algebraic conditions. Acta Math. Hungar. 53 (1989), 339–346. DOI 10.1007/BF01953371 | MR 1014917 | Zbl 0705.16021
[4] Dhara B.: Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings. Beitr. Algebra Geom. 53 (2012), no. 1, 203–209. DOI 10.1007/s13366-011-0051-9 | MR 2890375 | Zbl 1242.16039
[5] Chuang C.L.: GPIs having coefficients in Utumi quotient rings. Proc. Amer. Math. Soc. 103 (1988), no. 3, 723–728. DOI 10.1090/S0002-9939-1988-0947646-4 | MR 0947646 | Zbl 0656.16006
[6] Chuang C.L.: Differential identities with automorphism and anti-automorphism-II. J. Algebra 160 (1993), 291–335. DOI 10.1006/jabr.1993.1181 | MR 1237081
[7] Chuang C.L., Lee T.K.: Identities with a single skew derivation. J. Algebra 288 (2005), 59–77. DOI 10.1016/j.jalgebra.2003.12.032 | MR 2138371 | Zbl 1073.16021
[8] Eremita D., Ilisvic D.: On (anti-) multiplicative generalized derivations. Glas. Mat. Ser. III 47 (2012) no. 67, 105–118. DOI 10.3336/gm.47.1.08 | MR 2942778
[9] Erickson T.S., Martindale W.S.3rd., Osborn J.M.: Prime nonassociative algebras. Pacific. J. Math. 60 (1975), 49–63. DOI 10.2140/pjm.1975.60.49 | MR 0382379 | Zbl 0355.17005
[10] Gusic I.: A note on generalized derivations of prime rings. Glas. Mat. 40 (2005), 47–49. DOI 10.3336/gm.40.1.05 | MR 2195859 | Zbl 1072.16031
[11] Jacobson N.: Structure of Rings. Amer. Math. Soc. Colloq. Pub., 37, Providence, Rhode Island, 1964. MR 0222106 | Zbl 0098.25901
[12] Kharchenko V.K., Popov A.Z.: Skew derivations of prime rings. Comm. Algebra 20 (1992), 3321–3345. DOI 10.1080/00927879208824517 | MR 1186710 | Zbl 0783.16012
[13] Kharchenko V.K.: Generalized identities with automorphisms. Algebra i Logika 14 (1975), no. 2, 132–148. DOI 10.1007/BF01668425 | MR 0399153 | Zbl 0382.16009
[14] Lanski C.: An Engel condition with derivation. Proc. Amer. Math. Soc. 118 (1993), 75–80. DOI 10.1090/S0002-9939-1993-1132851-9 | MR 1132851 | Zbl 0869.16027
[15] Lee P.H., Wong T.L.: Derivations cocentralizing Lie ideals. Bull. Inst. Math. Acad. Sin. 23 (1995), no. 1–5. MR 1319474 | Zbl 0827.16025
[16] Rehman N.: On generalized derivations as homomorphism and anti-homomorphism. Glas. Mat. 39 (2014), 27–30. DOI 10.3336/gm.39.1.03 | MR 2055383
[17] Rehman N., Raza M.A.: On ideals with skew derivations of prime rings. Miskolc Math. Notes 15 (2014), no. 2, 717-724. MR 3302354 | Zbl 1324.16048
[18] Rehman N., Raza M.A.: On $m$-commuting mappings with skew derivations in prime rings. Algebra i Analiz 27 (2015) no. 4, 74–86. Zbl 1342.16040
[19] Rehman N., Raza M.A.: Generalized derivations as homomorphism and anti-homomorphism on Lie ideals. Arab. Math. J., http://dx.doi.org/10.1016/j.ajmsc.2014.09.001 DOI 10.1016/j.ajmsc.2014.09.001
[20] Wang Y., You H.: Derivations as homomorphism or anti-homomorphism on Lie ideals. Acta. Math. Sinica 23 (2007), 1149-1152. DOI 10.1007/s10114-005-0840-x | MR 2319944

Partner of