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A classification of generalized skew-derivations on multilinear polynomials in prime rings

Yıl 2025, Cilt: 54 Sayı: 3, 984 - 997, 24.06.2025

Basudeb Dhara , Gurninder S. Sandhu , Nripendu Bera

https://doi.org/10.15672/hujms.1472753

Öz

In this article, we are intended to examine generalized skew-derivations that act as Jordan homoderivations on multilinear polynomials in prime rings. More specifically, we show that if $F$ is generalized skew-derivation of a prime ring $R$ with associated automorphism $\alpha$ such that the relation $$F(X^2)=F(X)^2+F(X)X+XF(X)$$ holds for all $X\in f(R)$, where $f(x_1,\ldots,x_n)$ is a noncentral valued multilinear polynomial over extended centroid $C$, then either $F=0$ or $F=-id_{R}$ or $F=-id_{R}+\alpha$ (where $id_{R}$ denotes the identity map of $R$).

Anahtar Kelimeler

Derivation generalized derivation generalized skew derivation prime ring

Proje Numarası

This work is supported by a grant from Science and Engineering Research Board (SERB), DST, New Delhi, India. Grant No. is EMR/2016/004043 dated 29-Nov-2016

Kaynakça

  • [1] E. Albaş and N. Argaç, Generalized derivations of prime rings, Algebra Colloq. 11, 399-410, 2004.
  • [2] A. Ali and D. Kumar, Generalized derivations as homomorphisms or as antihomomorphisms in a prime ring, Hacet. J. Math. Stat. 38, 17-20, 2009.
  • [3] A. Ali, N. Rehman and S. Ali, On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar. 101, 79-82, 2003.
  • [4] K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math. 196, Marcel Dekker, New York, 1996.
  • [5] H. E. Bell and L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53 (3-4), 339-346, 1989.
  • [6] N. Bera and B. Dhara, Jordan Homoderivation behavior of generalized derivations in prime rings, Ukrainian Math. J. 75 (9), 1178-1194, 2023.
  • [7] J. C. Chang, Generalized skew derivations with annihilating Engel conditions, Taiwanese J. Math. 12, 1641-1650, 2008.
  • [8] J. C. Chang, Generalized skew derivations with nilpotent values on Lie ideals, Monatsh. Math. 161, 155-160, 2010.
  • [9] J. C. Chang, On the identity $h(x)=af(x)+g(x)b$, Taiwanese J. Math. 7 (1), 103-113, 2003.
  • [10] H. W. Cheng and F. Wei, Generalized skew derivations of rings, Adv. Math.(China) 35, 237-243, 2006.
  • [11] C. L. Chuang, Differential identities with automorphisms and antiautomorphisms I, J. Algebra 149, 371-404, 1992.
  • [12] C. L. Chuang, Differential identities with automorphisms and antiautomorphisms II, J. Algebra 160 (1), 130-171, 1993.
  • [13] C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (3), 723-728, 1988.
  • [14] C. L. Chuang and T. K. Lee, Identities with a single skew derivation, J. Algebra 288 (1), 59-77, 2005.
  • [15] M. M. El Sofy Aly, Rings with some kinds of mappings, M.Sc. Thesis, Cairo University, Branch of Fayoum, 2000.
  • [16] V. De Filippis, Generalized Derivations as Jordan Homomorphisms on Lie Ideals and Right Ideals, Acta Mathematica Sinica, 25 (12), 1965-1974, 2009.
  • [17] V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra 40, 1918-1932, 2012.
  • [18] V. De Filippis, B. Dhara and N. Bera, Generalized skew derivations and generalization of commuting maps on prime rings, Beitr. Algebra Geom. 63, 599-620, 2022.
  • [19] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Pub. 37, Amer. Math. Soc., Providence, RI, 1964.
  • [20] V. K. Kharchenko, Generalized identities with automorphisms, Algebra and Logic, 14, 132-148, 1975.
  • [21] C. Lanski, Differential identities, Lie ideals, and Posner’s theorem, Pacific J. Math. 134, 275-297, 1988.
  • [22] T. K. Lee, Generalized skew derivations characterized by acting on zero products, Pacific J. Math. 216, 293-301, 2004.
  • [23] T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1), 27-38, 1992.
  • [24] U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202, 97-103, 1975.
  • [25] K. S. Liu, Differential identities and constants of algebraic automorphisms in prime rings, Ph.D. Thesis, National Taiwan University 2006.
  • [26] W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12, 576-584, 1969.
  • [27] N. Rehman, On generalized derivations as homomorphisms and anti-homomorphisms, Glas. Mat. 39 (1), 27-30, 2004.
  • [28] G. Scudo, Generalized derivations acting as Lie homomorphisms on polynomials in prime rings, Southeast Asian Bull. Math. 38, 563-572, 2014.
  • [29] Y. Wang and H. You, Derivations as homomorphisms or anti-homomorphisms on Lie ideals, Acta Math. Sinica 32 (6), 1149-1152, 2007.
  • [30] T. L.Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq. 3, 369-478, 1996.
  • [31] X. Xu, J. Ma and F. Niu, Compositions, derivations and polynomials, Indian J. Pure Appl. Math. 44 (4), 543-556, 2013.

Yıl 2025, Cilt: 54 Sayı: 3, 984 - 997, 24.06.2025

Basudeb Dhara , Gurninder S. Sandhu , Nripendu Bera

https://doi.org/10.15672/hujms.1472753

Öz

Proje Numarası

This work is supported by a grant from Science and Engineering Research Board (SERB), DST, New Delhi, India. Grant No. is EMR/2016/004043 dated 29-Nov-2016

Kaynakça

  • [1] E. Albaş and N. Argaç, Generalized derivations of prime rings, Algebra Colloq. 11, 399-410, 2004.
  • [2] A. Ali and D. Kumar, Generalized derivations as homomorphisms or as antihomomorphisms in a prime ring, Hacet. J. Math. Stat. 38, 17-20, 2009.
  • [3] A. Ali, N. Rehman and S. Ali, On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar. 101, 79-82, 2003.
  • [4] K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math. 196, Marcel Dekker, New York, 1996.
  • [5] H. E. Bell and L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53 (3-4), 339-346, 1989.
  • [6] N. Bera and B. Dhara, Jordan Homoderivation behavior of generalized derivations in prime rings, Ukrainian Math. J. 75 (9), 1178-1194, 2023.
  • [7] J. C. Chang, Generalized skew derivations with annihilating Engel conditions, Taiwanese J. Math. 12, 1641-1650, 2008.
  • [8] J. C. Chang, Generalized skew derivations with nilpotent values on Lie ideals, Monatsh. Math. 161, 155-160, 2010.
  • [9] J. C. Chang, On the identity $h(x)=af(x)+g(x)b$, Taiwanese J. Math. 7 (1), 103-113, 2003.
  • [10] H. W. Cheng and F. Wei, Generalized skew derivations of rings, Adv. Math.(China) 35, 237-243, 2006.
  • [11] C. L. Chuang, Differential identities with automorphisms and antiautomorphisms I, J. Algebra 149, 371-404, 1992.
  • [12] C. L. Chuang, Differential identities with automorphisms and antiautomorphisms II, J. Algebra 160 (1), 130-171, 1993.
  • [13] C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (3), 723-728, 1988.
  • [14] C. L. Chuang and T. K. Lee, Identities with a single skew derivation, J. Algebra 288 (1), 59-77, 2005.
  • [15] M. M. El Sofy Aly, Rings with some kinds of mappings, M.Sc. Thesis, Cairo University, Branch of Fayoum, 2000.
  • [16] V. De Filippis, Generalized Derivations as Jordan Homomorphisms on Lie Ideals and Right Ideals, Acta Mathematica Sinica, 25 (12), 1965-1974, 2009.
  • [17] V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra 40, 1918-1932, 2012.
  • [18] V. De Filippis, B. Dhara and N. Bera, Generalized skew derivations and generalization of commuting maps on prime rings, Beitr. Algebra Geom. 63, 599-620, 2022.
  • [19] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Pub. 37, Amer. Math. Soc., Providence, RI, 1964.
  • [20] V. K. Kharchenko, Generalized identities with automorphisms, Algebra and Logic, 14, 132-148, 1975.
  • [21] C. Lanski, Differential identities, Lie ideals, and Posner’s theorem, Pacific J. Math. 134, 275-297, 1988.
  • [22] T. K. Lee, Generalized skew derivations characterized by acting on zero products, Pacific J. Math. 216, 293-301, 2004.
  • [23] T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1), 27-38, 1992.
  • [24] U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202, 97-103, 1975.
  • [25] K. S. Liu, Differential identities and constants of algebraic automorphisms in prime rings, Ph.D. Thesis, National Taiwan University 2006.
  • [26] W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12, 576-584, 1969.
  • [27] N. Rehman, On generalized derivations as homomorphisms and anti-homomorphisms, Glas. Mat. 39 (1), 27-30, 2004.
  • [28] G. Scudo, Generalized derivations acting as Lie homomorphisms on polynomials in prime rings, Southeast Asian Bull. Math. 38, 563-572, 2014.
  • [29] Y. Wang and H. You, Derivations as homomorphisms or anti-homomorphisms on Lie ideals, Acta Math. Sinica 32 (6), 1149-1152, 2007.
  • [30] T. L.Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq. 3, 369-478, 1996.
  • [31] X. Xu, J. Ma and F. Niu, Compositions, derivations and polynomials, Indian J. Pure Appl. Math. 44 (4), 543-556, 2013.
Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Cebir ve Sayı Teorisi
Bölüm Matematik
Yazarlar

Basudeb Dhara 0000-0002-8345-1362

Gurninder S. Sandhu 0000-0001-8618-6325

Nripendu Bera 0000-0002-6972-2568

Proje Numarası This work is supported by a grant from Science and Engineering Research Board (SERB), DST, New Delhi, India. Grant No. is EMR/2016/004043 dated 29-Nov-2016
Erken Görünüm Tarihi 27 Ocak 2025
Yayımlanma Tarihi 24 Haziran 2025
Gönderilme Tarihi 24 Nisan 2024
Kabul Tarihi 5 Ekim 2024
Yayımlandığı Sayı Yıl 2025 Cilt: 54 Sayı: 3

Kaynak Göster

APA Dhara, B., Sandhu, G. S., & Bera, N. (2025). A classification of generalized skew-derivations on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics, 54(3), 984-997. https://doi.org/10.15672/hujms.1472753
AMA Dhara B, Sandhu GS, Bera N. A classification of generalized skew-derivations on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics. Haziran 2025;54(3):984-997. doi:10.15672/hujms.1472753
Chicago Dhara, Basudeb, Gurninder S. Sandhu, ve Nripendu Bera. “A Classification of Generalized Skew-Derivations on Multilinear Polynomials in Prime Rings”. Hacettepe Journal of Mathematics and Statistics 54, sy. 3 (Haziran 2025): 984-97. https://doi.org/10.15672/hujms.1472753.
EndNote Dhara B, Sandhu GS, Bera N (01 Haziran 2025) A classification of generalized skew-derivations on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics 54 3 984–997.
IEEE B. Dhara, G. S. Sandhu, ve N. Bera, “A classification of generalized skew-derivations on multilinear polynomials in prime rings”, Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 3, ss. 984–997, 2025, doi: 10.15672/hujms.1472753.
ISNAD Dhara, Basudeb vd. “A Classification of Generalized Skew-Derivations on Multilinear Polynomials in Prime Rings”. Hacettepe Journal of Mathematics and Statistics 54/3 (Haziran 2025), 984-997. https://doi.org/10.15672/hujms.1472753.
JAMA Dhara B, Sandhu GS, Bera N. A classification of generalized skew-derivations on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics. 2025;54:984–997.
MLA Dhara, Basudeb vd. “A Classification of Generalized Skew-Derivations on Multilinear Polynomials in Prime Rings”. Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 3, 2025, ss. 984-97, doi:10.15672/hujms.1472753.
Vancouver Dhara B, Sandhu GS, Bera N. A classification of generalized skew-derivations on multilinear polynomials in prime rings. Hacettepe Journal of Mathematics and Statistics. 2025;54(3):984-97.

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