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Semiprime Rings with Multiplicative Generalized Derivations on Lie Ideals

Year 2017, , 473 - 479, 30.09.2017

Emine Koç

https://doi.org/10.17776/csj.340488

Abstract

 Let  be a torsion free semiprime ring. In [10], a map   is called a multiplicative
generalized derivation if there exists a map  such that  for all . Let  be a noncentral square-closed Lie
ideal of  and  multiplicative generalized
derivations associated to the maps of respectively such that and    for all
In the present paper, we shall prove that  is commuting map on  if any one of the following
holds: i)  ii)  If any one of the
conditions iii)  and iv) for all are satisfied, 

Keywords

Semiprime ring multiplicative generalized derivation generalized derivation Lie ideal

References

  • [1]. Ali, A., Yasen, M., and Anwar, M., Strong commutativity preserving mappings on semiprime rings, Bull. Korean Math. Soc., 43(4), 711-713, 2006.
  • [2]. Ashraf, M., Rehman, N. On derivations and commutativity in prime rings, East-West J. Math. 3(1), 87-91, 2001.
  • [3]. Ashraf, M., Asma, A. and Shakir, A. Some commutativiy theorems for rings with generalized derivations, Southeast Asain Bull. of Math. 31, 2007, 415-421.
  • [4]. Awtar, R., Lie structure in prime rings with derivations, Publ. Math. Debrecen , 31, 209-215, 1984.
  • [5]. Bell, H. E., Daif, M. N., On commutativity and strong commutativity preserving maps, Canad. Math. Bull., 37(4), 443-447, 1994.
  • [6]. Bergen, J., Herstein, I. N. and Kerr, W., Lie ideals and derivation of prime rings, J. of Algebra, 71, 259-267, 1981.
  • [7]. Bresar, M., On the distance of the compositions of two derivations to the generalized derivations, Glasgow Math. J., 33 (1), 89-93, 1991.
  • [8]. Daif, M. N., When is a multiplicative derivation additive, Int. J. Math. Math. Sci., 14(3), 615-618, 1991.
  • [9]. Daif, M. N., Tamman El-Sayiad, M. S., Multiplicative generalized derivation which are additive, East-West J. Math., 9(1), 31-37, 1997.
  • [10]. Dhara, B., Ali, S., On multiplicative (generalized) derivation in prime and semiprime rings, Aequat. Math., 86, 65-79, 2013.
  • [11]. Gölbaşı, Ö., Multiplicative generalized derivations on ideals in semiprime rings, Math. Slovaca, 66(6), 2016.
  • [12]. Goldman, H., Semrl, P., Multiplicative derivations on Monatsh Math., 121(3), 189-197, 1969.
  • [13]. Hongan, M., Rehman, N. and Al-Omary, R. M., Lie ideals and Jordan triple derivations in rings, Rend. Semin. Mat. Univ. Padova, 125, 147–156, 2011.
  • [14]. Martindale III, W. S., When are multiplicative maps additive, Proc. Amer. Math. Soc., 21, 695-698, 1969.
  • [15]. Posner, E. C., Derivations in prime rings, Proc Amer.Math.Soc., 8, 1093-1100, 1957.
  • [16]. Rehman, N., Hongan, M., Generalized Jordan derivations on Lie ideals associate with Hochschild cocycles of rings, Rend. Circ. Mat. Palermo, 60(3), 437-444, 2011.
  • [17]. Tiwari, S. K., Sharma, R. K. and Dhara, B., Multiplicative (generalized)-derivation in semiprime rings, Beitr. Algebra Geom., 1–15, 2015.

Lie İdealler Üzerinde Çarpımsal Genelleştirilmiş Türevli Yarıasal Halkalar

Year 2017, , 473 - 479, 30.09.2017

Emine Koç

https://doi.org/10.17776/csj.340488

Abstract

R, 2- torsion free bir yarıasal halka olsun. [10] dan,
eğer her  için  koşulunu sağlayan bir  dönüşümü varsa  dönüşümüne  halkasının  ile belirlenmiş bir çarpımsal genelleştirilmiş
türevi denir.   halkasının
bir merkez tarafından
kapsanılmayan kare-kapalı Lie ideali,  dönüşümleri
 halkasının sırasıyla  dönüşümleri ile belirlenmiş çarpımsal
genelleştirilmiş türevleri, ve her   olsun. Bu çalışmada, aşağıdaki
koşullardan biri sağlanırsa  üzerinde komüting dönüşüm olduğu
gösterilecektir: i)  ii)  Ayrıca her  için iii)  iv)  koşullarından biri
sağlanırsa bu durumda olduğu ispatlanacaktır. 

Keywords

Yarıasal halka çarpımsal genelleştirilmiş türev genelleştirilmiş türev Lie ideal

References

  • [1]. Ali, A., Yasen, M., and Anwar, M., Strong commutativity preserving mappings on semiprime rings, Bull. Korean Math. Soc., 43(4), 711-713, 2006.
  • [2]. Ashraf, M., Rehman, N. On derivations and commutativity in prime rings, East-West J. Math. 3(1), 87-91, 2001.
  • [3]. Ashraf, M., Asma, A. and Shakir, A. Some commutativiy theorems for rings with generalized derivations, Southeast Asain Bull. of Math. 31, 2007, 415-421.
  • [4]. Awtar, R., Lie structure in prime rings with derivations, Publ. Math. Debrecen , 31, 209-215, 1984.
  • [5]. Bell, H. E., Daif, M. N., On commutativity and strong commutativity preserving maps, Canad. Math. Bull., 37(4), 443-447, 1994.
  • [6]. Bergen, J., Herstein, I. N. and Kerr, W., Lie ideals and derivation of prime rings, J. of Algebra, 71, 259-267, 1981.
  • [7]. Bresar, M., On the distance of the compositions of two derivations to the generalized derivations, Glasgow Math. J., 33 (1), 89-93, 1991.
  • [8]. Daif, M. N., When is a multiplicative derivation additive, Int. J. Math. Math. Sci., 14(3), 615-618, 1991.
  • [9]. Daif, M. N., Tamman El-Sayiad, M. S., Multiplicative generalized derivation which are additive, East-West J. Math., 9(1), 31-37, 1997.
  • [10]. Dhara, B., Ali, S., On multiplicative (generalized) derivation in prime and semiprime rings, Aequat. Math., 86, 65-79, 2013.
  • [11]. Gölbaşı, Ö., Multiplicative generalized derivations on ideals in semiprime rings, Math. Slovaca, 66(6), 2016.
  • [12]. Goldman, H., Semrl, P., Multiplicative derivations on Monatsh Math., 121(3), 189-197, 1969.
  • [13]. Hongan, M., Rehman, N. and Al-Omary, R. M., Lie ideals and Jordan triple derivations in rings, Rend. Semin. Mat. Univ. Padova, 125, 147–156, 2011.
  • [14]. Martindale III, W. S., When are multiplicative maps additive, Proc. Amer. Math. Soc., 21, 695-698, 1969.
  • [15]. Posner, E. C., Derivations in prime rings, Proc Amer.Math.Soc., 8, 1093-1100, 1957.
  • [16]. Rehman, N., Hongan, M., Generalized Jordan derivations on Lie ideals associate with Hochschild cocycles of rings, Rend. Circ. Mat. Palermo, 60(3), 437-444, 2011.
  • [17]. Tiwari, S. K., Sharma, R. K. and Dhara, B., Multiplicative (generalized)-derivation in semiprime rings, Beitr. Algebra Geom., 1–15, 2015.
There are 17 citations in total.

Details

Journal Section Articles
Authors

Emine Koç

Publication Date September 30, 2017
Submission Date May 15, 2017
Acceptance Date June 5, 2017
Published in Issue Year 2017

Cite

APA Koç, E. (2017). Semiprime Rings with Multiplicative Generalized Derivations on Lie Ideals. Cumhuriyet Science Journal, 38(3), 473-479. https://doi.org/10.17776/csj.340488