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Additive maps on prime and semiprime rings with involution

Year 2020, Volume: 49 Issue: 3, 1126 - 1133, 02.06.2020
https://doi.org/10.15672/hujms.661178

Abstract

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.

References

  • [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.
Year 2020, Volume: 49 Issue: 3, 1126 - 1133, 02.06.2020
https://doi.org/10.15672/hujms.661178

Abstract

References

  • [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.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

A. Alahmadi This is me 0000-0002-7758-3537

H. Alhazmi This is me 0000-0001-7190-5884

Shakir Ali This is me 0000-0001-5162-7522

Nadeem Dar 0000-0003-0074-2912

Abdul Khan

Publication Date June 2, 2020
Published in Issue Year 2020 Volume: 49 Issue: 3

Cite

APA Alahmadi, A., Alhazmi, H., Ali, S., Dar, N., et al. (2020). Additive maps on prime and semiprime rings with involution. Hacettepe Journal of Mathematics and Statistics, 49(3), 1126-1133. https://doi.org/10.15672/hujms.661178
AMA Alahmadi A, Alhazmi H, Ali S, Dar N, Khan A. Additive maps on prime and semiprime rings with involution. Hacettepe Journal of Mathematics and Statistics. June 2020;49(3):1126-1133. doi:10.15672/hujms.661178
Chicago Alahmadi, A., H. Alhazmi, Shakir Ali, Nadeem Dar, and Abdul Khan. “Additive Maps on Prime and Semiprime Rings With Involution”. Hacettepe Journal of Mathematics and Statistics 49, no. 3 (June 2020): 1126-33. https://doi.org/10.15672/hujms.661178.
EndNote Alahmadi A, Alhazmi H, Ali S, Dar N, Khan A (June 1, 2020) Additive maps on prime and semiprime rings with involution. Hacettepe Journal of Mathematics and Statistics 49 3 1126–1133.
IEEE A. Alahmadi, H. Alhazmi, S. Ali, N. Dar, and A. Khan, “Additive maps on prime and semiprime rings with involution”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 3, pp. 1126–1133, 2020, doi: 10.15672/hujms.661178.
ISNAD Alahmadi, A. et al. “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.
JAMA Alahmadi A, Alhazmi H, Ali S, Dar N, Khan A. Additive maps on prime and semiprime rings with involution. Hacettepe Journal of Mathematics and Statistics. 2020;49:1126–1133.
MLA Alahmadi, A. et al. “Additive Maps on Prime and Semiprime Rings With Involution”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 3, 2020, pp. 1126-33, doi:10.15672/hujms.661178.
Vancouver Alahmadi A, Alhazmi H, Ali S, Dar N, Khan A. Additive maps on prime and semiprime rings with involution. Hacettepe Journal of Mathematics and Statistics. 2020;49(3):1126-33.