Generalized Derivations Acting on Multilinear Polynomials in Prime Rings and Banach Algebras

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• 作者：Basudeb Dhara ; Nurcan Argaç
• 关键词：Prime ring ; Derivation ; Generalized derivation ; Extended centroid ; Utumi quotient ring ; Banach algebra
• 刊名：Communications in Mathematics and Statistics
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：4
• 期：1
• 页码：39-54
• 全文大小：440 KB
• 参考文献：1.Argaç, N., De Filippis, V.: Actions of generalized derivations on multilinear polynomials in prime rings. Algebra Colloq. 18(Spec 01), 955–964 (2011)MathSciNet CrossRef MATH
  2.Argaç, N., Demir, Ç.: Generalized derivations of prime rings on multilinear polynomials with annhilator conditions. Turk. J. Math. 37(2), 231–243 (2013)MATH
  3.Demir, Ç., Argaç, N.: Prime rings with generalized derivations on right ideals. Algebra Colloq. 18(1), 987–908 (2011)MathSciNet CrossRef MATH
  4.Beidar, K.I.: Rings of quotients of semiprime rings. Vestn. Mosk. Univ. Ser. I Mat. Mek. (Engl. Transl: Moskow Univ. Math. Bull.) 33, 36–42 (1978)
  5.Bergen, J., Herstein, I.N., Kerr, J.W.: Lie ideals and derivations of prime rings. J. Algebra 71, 259–267 (1981)MathSciNet CrossRef MATH
  6.Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Pure and Applied Mathematics, vol. 196. Marcel Dekker, New York (1996)
  7.Carini, L., De Filippis, V.: Centralizers of generalized derivations on multilinear polynomials in prime rings. Sib. Math. J. 53(6), 1051–1060 (2012)MathSciNet CrossRef MATH
  8.Chen, H.Y.: Generalized derivations with Engel conditions on polynomials. Comm. Algebra 39, 3709–3721 (2011)MathSciNet CrossRef MATH
  9.Chuang, C.L.: GPIs having coefficients in Utumi quotient rings. Proc. Amer. Math. Soc. 103(3), 723–728 (1988)MathSciNet CrossRef MATH
  10.Chuang, C.L.: The additive subgroup generated by a polynomial. Isr. J. Math. 59(1), 98–106 (1987)MathSciNet CrossRef MATH
  11.Chuang, C.L.: Hypercentral derivations. J. Algebra 166(1), 34–71 (1994)MathSciNet CrossRef MATH
  12.De Filippis, V., Di Vincenzo, O.M.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Comm. Algebra 40, 1918–1932 (2012)MathSciNet CrossRef MATH
  13.Dhara, B., Argaç, N., Pradhan, K.G.: Annihilator condition of a pair of derivations in prime and semiprime rings. Indian J. Pure Appl. Math. (to appear)
  14.Erickson, T.S., Martindale III, W.S., Osborn, J.M.: Prime nonassociative algebras. Pacific J. Math. 60, 49–63 (1975)MathSciNet CrossRef MATH
  15.Faith, C., Utumi, Y.: On a new proof of Litoff’s theorem. Acta Math. Acad. Sci. Hung. 14, 369–371 (1963)MathSciNet CrossRef MATH
  16.Jacobson, N.: Structure of Rings, vol. 37. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence (1964)
  17.Johnson, B.E., Sinclair, A.M.: Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90, 1067–1073 (1968)MathSciNet CrossRef MATH
  18.Kharchenko, V.K.: Differential identity of prime rings. Algebra Log. 17, 155–168 (1978)CrossRef MATH
  19.Lee, T.K.: Generalized derivations of left faithful rings. Comm. Algebra 27(8), 4057–4073 (1999)MathSciNet CrossRef MATH
  20.Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sin. 20(1), 27–38 (1992)MathSciNet MATH
  21.Leron, U.: Nil and power central polynomials in rings. Trans. Amer. Math. Soc. 202, 97–103 (1975)MathSciNet CrossRef MATH
  22.Martindale III, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969)MathSciNet CrossRef MATH
  23.Posner, E.C.: Derivations in prime rings. Proc. Amer. Math. Soc. 8, 1093–1100 (1957)MathSciNet CrossRef MATH
  24.Rania, F.: Generalized derivations and annihilator conditions in prime rings. Int. J. Algebra 2(20), 963–969 (2008)MathSciNet MATH
  25.Sinclair, A.M.: Jordan homomorphisms and derivations on semisimple Banach algebras. Proc. Amer. Math. Soc. 24, 209–214 (1970)MathSciNet MATH
  
• 作者单位：Basudeb Dhara (1)
  Nurcan Argaç (2)
  
  1. Department of Mathematics, Belda College, Belda, Paschim Medinipur, W.B., 721424, India
  2. Department of Mathematics, Science Faculty, Ege University, 35100, Bornova, Izmir, Turkey
  
• 刊物主题：Mathematics, general; Statistics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：2194-671X

文摘

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F and G, the two nonzero 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 $$\begin{aligned} F(G(f(x_1,\ldots ,x_n))f(x_1,\ldots ,x_n))=0 \end{aligned}$$for all \(x_1,\ldots ,x_n \in I\), then one of the followings holds: (1) there exist \(a,b\in U\) such that \(F(x)=ax\) and \(G(x)=bx\) for all \(x\in R\) with \(ab=0\); (2) there exist \(a,b,p\in U\) such that \(F(x)=ax+xb\) and \(G(x)=px\) for all \(x\in R\) with \(F(p)=0\) and \(f(x_1,\ldots ,x_n)^2\) is central valued on R. We also obtain some related results in cases where R is a semiprime ring and Banach algebra. Keywords Prime ring Derivation Generalized derivation Extended centroid Utumi quotient ring Banach algebra Mathematics Subject Classification 16W25 16N60 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (25) References1.Argaç, N., De Filippis, V.: Actions of generalized derivations on multilinear polynomials in prime rings. Algebra Colloq. 18(Spec 01), 955–964 (2011)MathSciNetCrossRefMATH2.Argaç, N., Demir, Ç.: Generalized derivations of prime rings on multilinear polynomials with annhilator conditions. Turk. J. Math. 37(2), 231–243 (2013)MATH3.Demir, Ç., Argaç, N.: Prime rings with generalized derivations on right ideals. Algebra Colloq. 18(1), 987–908 (2011)MathSciNetCrossRefMATH4.Beidar, K.I.: Rings of quotients of semiprime rings. Vestn. Mosk. Univ. Ser. I Mat. Mek. (Engl. Transl: Moskow Univ. Math. Bull.) 33, 36–42 (1978)5.Bergen, J., Herstein, I.N., Kerr, J.W.: Lie ideals and derivations of prime rings. J. Algebra 71, 259–267 (1981)MathSciNetCrossRefMATH6.Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Pure and Applied Mathematics, vol. 196. Marcel Dekker, New York (1996)7.Carini, L., De Filippis, V.: Centralizers of generalized derivations on multilinear polynomials in prime rings. Sib. Math. J. 53(6), 1051–1060 (2012)MathSciNetCrossRefMATH8.Chen, H.Y.: Generalized derivations with Engel conditions on polynomials. Comm. Algebra 39, 3709–3721 (2011)MathSciNetCrossRefMATH9.Chuang, C.L.: GPIs having coefficients in Utumi quotient rings. Proc. Amer. Math. Soc. 103(3), 723–728 (1988)MathSciNetCrossRefMATH10.Chuang, C.L.: The additive subgroup generated by a polynomial. Isr. J. Math. 59(1), 98–106 (1987)MathSciNetCrossRefMATH11.Chuang, C.L.: Hypercentral derivations. J. Algebra 166(1), 34–71 (1994)MathSciNetCrossRefMATH12.De Filippis, V., Di Vincenzo, O.M.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Comm. Algebra 40, 1918–1932 (2012)MathSciNetCrossRefMATH13.Dhara, B., Argaç, N., Pradhan, K.G.: Annihilator condition of a pair of derivations in prime and semiprime rings. Indian J. Pure Appl. Math. (to appear)14.Erickson, T.S., Martindale III, W.S., Osborn, J.M.: Prime nonassociative algebras. Pacific J. Math. 60, 49–63 (1975)MathSciNetCrossRefMATH15.Faith, C., Utumi, Y.: On a new proof of Litoff’s theorem. Acta Math. Acad. Sci. Hung. 14, 369–371 (1963)MathSciNetCrossRefMATH16.Jacobson, N.: Structure of Rings, vol. 37. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence (1964)17.Johnson, B.E., Sinclair, A.M.: Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90, 1067–1073 (1968)MathSciNetCrossRefMATH18.Kharchenko, V.K.: Differential identity of prime rings. Algebra Log. 17, 155–168 (1978)CrossRefMATH19.Lee, T.K.: Generalized derivations of left faithful rings. Comm. Algebra 27(8), 4057–4073 (1999)MathSciNetCrossRefMATH20.Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sin. 20(1), 27–38 (1992)MathSciNetMATH21.Leron, U.: Nil and power central polynomials in rings. Trans. Amer. Math. Soc. 202, 97–103 (1975)MathSciNetCrossRefMATH22.Martindale III, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969)MathSciNetCrossRefMATH23.Posner, E.C.: Derivations in prime rings. Proc. Amer. Math. Soc. 8, 1093–1100 (1957)MathSciNetCrossRefMATH24.Rania, F.: Generalized derivations and annihilator conditions in prime rings. Int. J. Algebra 2(20), 963–969 (2008)MathSciNetMATH25.Sinclair, A.M.: Jordan homomorphisms and derivations on semisimple Banach algebras. Proc. Amer. Math. Soc. 24, 209–214 (1970)MathSciNetMATH About this Article Title Generalized Derivations Acting on Multilinear Polynomials in Prime Rings and Banach Algebras Journal Communications in Mathematics and Statistics Volume 4, Issue 1 , pp 39-54 Cover Date2016-03 DOI 10.1007/s40304-015-0073-y Print ISSN 2194-6701 Online ISSN 2194-671X Publisher Springer Berlin Heidelberg Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Statistics, general Keywords Prime ring Derivation Generalized derivation Extended centroid Utumi quotient ring Banach algebra 16W25 16N60 Authors Basudeb Dhara (1) Nurcan Argaç (2) Author Affiliations 1. Department of Mathematics, Belda College, Belda, Paschim Medinipur, W.B., 721424, India 2. Department of Mathematics, Science Faculty, Ege University, 35100, Bornova, Izmir, Turkey Continue reading... To view the rest of this content please follow the download PDF link above.