A note on semiderivations in prime rings and C*-algebras

Автор: Raza Mohd Arif, Rehman Nadeem Ur

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 2 т.23, 2021 года.

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Let R be a prime ring with the extended centroid C and the Matrindale quotient ring Q. An additive mapping F:R→R is called a semiderivation associated with a mapping G:R→R, whenever F(xy)=F(x)G(y)+xF(y)=F(x)y+G(x)F(y) and F(G(x))=G(F(x)) holds for all x,y∈R. In this manuscript, we investigate and describe the structure of a prime ring R which satisfies F(xm∘yn)∈Z(R) for all x,y∈R, where m,n∈Z+ and F:R→R is a semiderivation with an automorphism ξ of R. Further, as an application of our ring theoretic results, we discussed the nature of C∗-algebras. To be more specific, we obtain for any primitive C∗-algebra A. If an anti-automorphism ζ:A→A satisfies the relation (xn)ζ+xn∗∈Z(A) for every x,y∈A, then A is C∗-W4-algebra, i.e., A satisfies the standard identity W4(a1,a2,a3,a4)=0 for all a1,a2,a3,a4∈A.

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Prime ring, automorphism, semiderivation

Короткий адрес: https://sciup.org/143175704

IDR: 143175704   |   DOI: 10.46698/d4945-5026-4001-v

Список литературы A note on semiderivations in prime rings and C*-algebras

  • Beidar, K. I., Martindale III, W. S. and Mikhalev, V. Rings with Generalized Identities, Pure and Applied Math., vol. 196, New York, Dekker, 1996.
  • Bergen, J. Derivations in Prime Ring, Canadian Mathematical Bulletin, 1983, vol. 26, no. 3, pp. 267–270. DOI: 10.4153/CMB-1983-042-2.
  • Bre˘sar, M. Semiderivations of Prime Rings, Proceedings of the American Mathematical Society, 1990, vol. 108, no. 4, pp. 859–860. DOI: 10.1090/S0002-9939-1990-1007488-X.
  • Posner, E. C. Derivations in Prime Rings, Proceedings of the American Mathematical Society, 1957, vol. 8, no. 6, pp. 1093–1100. DOI: 10.1090/S0002-9939-1957-0095863-0.
  • Lanski, C. Differential Identities, Lie Ideals and Posner’s Theorems, Pacific Journal of Mathematics, 1988, vol. 134, no. 2, pp. 275–297. DOI: 10.2140/pjm.1988.134.275.
  • Daif, M. N. and Bell, H. E. Remarks on Derivations on Semiprime Rings, International Journal of Mathematics and Mathematical Sciences, 1992, vol. 15, Article ID 863506, 2 p. DOI: 10.1155/S0161171292000255.
  • Ashraf, M. and Rehman, N. On Commutativity of Rings with Derivations, Results in Mathematics, 2002, vol. 42, no. 1–2, pp. 3–8. DOI: 10.1007/BF03323547.
  • Herstein, I. N. A Remark on Rings and Algebras, Michigan Mathematical Journal, 1963, vol. 10, no. 3, pp. 269–272. DOI: 10.1307/mmj/1028998910.
  • Bell, H. E. On the Commutativity of Prime Rings with Derivation, Quaestiones Mathematicae, 1999, vol. 22, pp. 329–335. DOI: 10.1080/16073606.1999.9632085.
  • Ali, S., Khan, M. S., Khan, A. N. and Muthana, N. M. On Rings and Algebras with Derivations, Journal of Algebra and its Applications, 2016, vol. 15, no. 6, 1650107, 10 p. DOI: 10.1142/S0219498816500225.
  • Ali, S., Ashraf, M., Raza, M. A. and Khan, A. N. n-Commuting Mappings on (Semi)-Prime Rings with Application, Communications in Algebra, 2019, vol. 47, no. 5, pp. 2262–2270. DOI: 10.1080/00927872.2018.1536203.
  • Raza, M. A. and Rehman, N. An Identity on Automorphisms of Lie Ideals in Prime Rings, Annali dell’Universita’ di Ferrara, 2016, vol. 62, no. 1, pp. 143–150. DOI: 10.1007/s11565-016-0240-4.
  • Raza, M. A. and Rehman, N. On Prime and Semiprime Rings with Generalized Derivations and Non- Commutative Banach Algebras, Proceedings–Mathematical Sciences, 2016, vol. 126, no. 3, pp. 389–398.
  • Rehman, N. and Raza, M. A. On m-Commuting Mappings with Skew Derivations in Prime Rings, St. Petersburg Mathematical Journal, 2016, vol. 27, no. 4, pp. 641–650. DOI: 10.1090/spmj/1411.
  • Huang, S. Semiderivations with Power Values on Lie Ideals in Prime Rings, Ukrainian Mathematical Journal, 2013, vol. 65, no. 6, pp. 967–971. DOI: 10.1007/s11253-013-0834-2.
  • Chuang, C. L. GPIs Having Quotients in Utumi Quotient Rings, Proceedings of the American Mathematical Society, 1988, vol. 103, no. 3, pp. 723–728. DOI: 10.1090/S0002-9939-1988-0947646-4.
  • Chuang, C. L. Differential Identities with Automorphism and Anti-Automorphism-I, Journal of Algebra, 1992, vol. 149, pp. 371–404. DOI: 10.1016/0021-8693(92)90023-F.
  • Bergen, J., Herstein, I. N. and Kerr, J.W. Lie Ideals and Derivations of Prime Rings, Journal of Algebra, 1981, vol. 71, pp. 259–267. DOI: 10.1016/0021-8693(81)90120-4.
  • Lanski, C. and Montgomery, S. Lie Structure of Prime Rings of Characteristic 2, Pacific Journal of Mathematics, 1972, vol. 42, no. 1, pp. 117–136. DOI: 10.2140/pjm.1972.42.117.
  • Jacobson, N. Structure of Rings, Amer. Math. Soc. Colloq. Pub., vol. 37, Amer. Math. Soc., Providence, RI, 1964.
  • Chuang, C. L., Chou, M. C. and Liu, C. K. Skew Derivations with Annihilating Engel Conditions, Publicationes Mathematicae Debrecen, 2006, vol. 68, no. 1–2, pp. 161–170.
  • Martindale 3rd, W. S. Prime Rings Satisfying a Generalized Polynomial Identity, Journal of Algebra, 1969, vol. 12, no. 4, pp. 576–584. DOI: 10.1016/0021-8693(69)90029-5.
  • Chuang, C. L. Differential Identities with Automorphisms and Antiautomorphisms, II, Journal of Algebra, 1993, vol. 160, no. 1, pp. 291–335. DOI: 10.1006/jabr.1993.1181.
  • Lee, T. K. Semiprime Rings with Differential Identities, Bulletin of the Institute of Mathematics Academia Sinica, 1992, vol. 20, no. 1, pp. 27–38.
  • Chuang, C. L. The Additive Subgroup Generated by a Polynomial, Israel Journal of Mathematics, 1987, vol. 59, no. 1, pp. 98–106. DOI: 10.1007/BF02779669.
  • Krupnik, N., Roch, S. and Silbermann, B. On C_-Algebras Generated by Idempotents, Journal of Functional Analysis, 1996, vol. 137, no. 2, pp. 303–319. DOI: 10.1006/jfan.1996.0048.
  • M¨uller, V. Nil, Nilpotent and PI-Algebras, Functional Analysis and Operator Theory, Banach Center Publications, 1994, vol. 30, pp. 259–265. DOI: 10.4064/-30-1-259-265.
  • Murphy, G. J. C_-Algebras and Operator Theory, New York, Academic Press Inc., 1990.
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