Year 2020, Volume 49 , Issue 3, Pages 1126 - 1133 2020-06-02

Additive maps on prime and semiprime rings with involution

A. ALAHMADİ [1] , H. ALHAZMİ [2] , Shakir ALİ [3] , Nadeem DAR [4] , Abdul KHAN [5]


Let $R$ be an associative ring. An additive map $x\mapsto x^*$ of $R$ into itself is called an involution if (i) $(xy)^*=y^*x^*$ and (ii) $(x^*)^*=x$ hold for all $x\in R$. The main purpose of this paper is to study some additive mappings on prime and semiprime rings with involution. Moreover, some examples are given to demonstrate that the restrictions imposed on the hypothesis of the various results are not superfluous.
prime ring, semiprime ring, normal ring, involution, generalized derivation, left $*$-centralizer, Jordan left $*$-centralizer, generalized derivation
  • [1] S. Ali, On generalized $*$-derivations in $*$-rings, Pales. J. Math. 1, 32–37, 2012.
  • [2] S. Ali and N.A. Dar, On $*$-centralizing mappings in rings with involution, Georgian Math. J. 21 (1), 25–28, 2014.
  • [3] S. Ali, N.A. Dar, and J. Vukman, Jordan left $*$-centralizers of prime and semiprime rings with involution, Beitr. Algebra Geom. 54, 609–624, 2013.
  • [4] K.I. Beidar, W.S. Martindale III, and A.V. Mikhalev, Rings with generalized identities, Dekker, New York-Basel-Hong Kong, 1996.
  • [5] H.E. Bell and W.S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1), 92–101, 1987.
  • [6] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156, 385–394, 1993.
  • [7] M. Bresar and J. Vukman, On some additive mappings in rings with involution, Aequationes Math. 38, 178–185, 1989.
  • [8] C.L. Chaung, $*$-differential identities of prime rings with involution, Tran. Amer. Math. Soc. 316 (1), 251–279, 1989.
  • [9] V. De Filippis, Posner’s second theorem and an annihilator condition with generalized derivations, Turkish J. Math. 32 (2), 197–211, 2008.
  • [10] V. De Filippis and M.S. Tammam El-Sayiad, A note on Posner’s theorem with generalized derivations on Lie ideals, Rend. Semin. Mat. Univ. Padova 122, 55–64, 2009.
  • [11] B. Dhara and S. Ali, On n-centralizing generalized derivations in semiprime rings with applications to $C^*$-algebras, J. Algebra Appl. 11 (6), 1250111, 2012.
  • [12] B. Dhara and V. De Filippis, Notes on generalized derivations on Lie ideals in prime rings, Bull. Korean Math. Soc. 46 (3), 599–605, 2009.
  • [13] M. Fošner and J. Vukman, A characterization of two-sided centralizers on prime rings, Taiwan J. Math. 11, 1431–1441, 2007.
  • [14] M. Fošner and J. Vukman, An equation related to two-sided centralizers in prime rings, Rocky Mountain J. Math. 41 (3), 765–776, 2011.
  • [15] O. Golbasi and E. Koc, Notes on commutativity of prime rings with generalized derivation, Commun. Fac. Sci. Univ. Ank. Ser. A1-Math. Stat. 58 (2), 39–46, 2009.
  • [16] I.N. Herstein, Jordan derivations on prime rings, Proc. Amer. Math. Soc. 8, 1104– 1110, 1957.
  • [17] I.N. Herstein, Rings with Involution, University of Chicago Press, Chicago, 1976.
  • [18] B. Hvala, Generalized derivations in rings, Comm. Algebra 26, 1147–1166, 1998.
  • [19] T.K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (8), 4057– 4073, 1999.
  • [20] F.A. Lopez, G.E. Rus, and S.E. Campos, Structure theorem for prime rings satisfying a generalized identities, Comm. Algebra 22 (5), 1729–1740, 1994.
  • [21] J. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19, 113– 117, 1976.
  • [22] J. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. 27, 122–126, 1984.
  • [23] W.S. Martindale III, Prime rings satisfying generalized polynomial identities, J. Algebra 12, 574–584, 1969.
  • [24] L. Molńar, On centralizers of an $H^*$-algebra, Publ. Math. Debrecen 46, 89–95, 1995.
  • [25] L. Oukhtite, A. Mamouni, Generalized derivations centralizing on Jordan ideals of rings with involution, Turkish J. Math. 38 (2), 225–232, 2014.
  • [26] L. Oukhtite, S. Salhi, and L. Taoufiq, Generalized derivations and commutativity of rings with involution, Beitr. Algebra Geom. 51 (2), 345–351, 2010.
  • [27] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 37, 27–28, 1988.
  • [28] N. Rehman and V. De Filippis, Commutativity and skew-commutativity conditions with generalized derivations, Algebra Colloq. 17, 841–850, 2010.
  • [29] I. Kosi-Ulbl and J. Vukman, On centralizers of standard operator algebras and semisimple $H^*$-algebras, Acta Math. Hung. 110, 217–223, 2006.
  • [30] J. Vukman, Centralizers in prime and semiprime rings, Comment. Math. Univ. Carolin. 38, 231–240, 1997.
  • [31] J. Vukman, Centralizers on semiprime rings, Comment. Math. Univ. Carolin. 42, 237–245, 2001.
  • [32] J. Vukman and I. Kosi-Ulbl, Centralizers on rings and algebras, Bull. Austral. Math. Soc. 71, 225–234, 2005.
  • [33] J. Vukman and I. Kosi-Ulbl, On centralizers of semiprime rings with involution, Stud. Sci. Math. Hungar. 43, 77–83, 2006.
  • [34] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32, 609–614, 1991.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0002-7758-3537
Author: A. ALAHMADİ
Institution: King Abdulaziz University
Country: Saudi Arabia


Orcid: 0000-0001-7190-5884
Author: H. ALHAZMİ
Institution: King Abdulaziz University
Country: Saudi Arabia


Orcid: 0000-0001-5162-7522
Author: Shakir ALİ
Institution: King Abdulaziz University
Country: Saudi Arabia


Orcid: 0000-0003-0074-2912
Author: Nadeem DAR
Institution: IUST
Country: India


Orcid: 0000-0001-5861-6137
Author: Abdul KHAN (Primary Author)
Institution: King Abdulaziz University
Country: Saudi Arabia


Dates

Publication Date : June 2, 2020

Bibtex @research article { hujms661178, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1126 - 1133}, doi = {10.15672/hujms.661178}, title = {Additive maps on prime and semiprime rings with involution}, key = {cite}, author = {Alahmadi̇, A. and Alhazmi̇, H. and Ali̇, Shakir and Dar, Nadeem and Khan, Abdul} }
APA Alahmadi̇, A , Alhazmi̇, H , Ali̇, S , Dar, N , Khan, A . (2020). Additive maps on prime and semiprime rings with involution . Hacettepe Journal of Mathematics and Statistics , 49 (3) , 1126-1133 . DOI: 10.15672/hujms.661178
MLA Alahmadi̇, A , Alhazmi̇, H , Ali̇, S , Dar, N , Khan, A . "Additive maps on prime and semiprime rings with involution" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1126-1133 <https://dergipark.org.tr/en/pub/hujms/issue/54699/661178>
Chicago Alahmadi̇, A , Alhazmi̇, H , Ali̇, S , Dar, N , Khan, A . "Additive maps on prime and semiprime rings with involution". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1126-1133
RIS TY - JOUR T1 - Additive maps on prime and semiprime rings with involution AU - A. Alahmadi̇ , H. Alhazmi̇ , Shakir Ali̇ , Nadeem Dar , Abdul Khan Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.661178 DO - 10.15672/hujms.661178 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1126 EP - 1133 VL - 49 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.661178 UR - https://doi.org/10.15672/hujms.661178 Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Additive maps on prime and semiprime rings with involution %A A. Alahmadi̇ , H. Alhazmi̇ , Shakir Ali̇ , Nadeem Dar , Abdul Khan %T Additive maps on prime and semiprime rings with involution %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 3 %R doi: 10.15672/hujms.661178 %U 10.15672/hujms.661178
ISNAD Alahmadi̇, A. , Alhazmi̇, H. , Ali̇, Shakir , Dar, Nadeem , Khan, Abdul . "Additive maps on prime and semiprime rings with involution". Hacettepe Journal of Mathematics and Statistics 49 / 3 (June 2020): 1126-1133 . https://doi.org/10.15672/hujms.661178
AMA Alahmadi̇ A , Alhazmi̇ H , Ali̇ S , Dar N , Khan A . Additive maps on prime and semiprime rings with involution. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 1126-1133.
Vancouver Alahmadi̇ A , Alhazmi̇ H , Ali̇ S , Dar N , Khan A . Additive maps on prime and semiprime rings with involution. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 1126-1133.