On left ( θ , ϕ ) -derivations of prime rings

Mohammad Ashraf

Archivum Mathematicum (2005)

  • Volume: 041, Issue: 2, page 157-166
  • ISSN: 0044-8753

Abstract

top
Let R be a 2 -torsion free prime ring. Suppose that θ , φ are automorphisms of R . In the present paper it is established that if R admits a nonzero Jordan left ( θ , θ ) -derivation, then R is commutative. Further, as an application of this resul it is shown that every Jordan left ( θ , θ ) -derivation on R is a left ( θ , θ ) -derivation on R . Finally, in case of an arbitrary prime ring it is proved that if R admits a left ( θ , φ ) -derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of R , then d = 0 on R .

How to cite

top

Ashraf, Mohammad. "On left $(\theta ,\varphi )$-derivations of prime rings." Archivum Mathematicum 041.2 (2005): 157-166. <http://eudml.org/doc/249503>.

@article{Ashraf2005,
abstract = {Let $R$ be a $2$-torsion free prime ring. Suppose that $\theta , \phi $ are automorphisms of $R$. In the present paper it is established that if $R$ admits a nonzero Jordan left $(\theta ,\theta )$-derivation, then $R$ is commutative. Further, as an application of this resul it is shown that every Jordan left $(\theta ,\theta )$-derivation on $R$ is a left $(\theta ,\theta )$-derivation on $R$. Finally, in case of an arbitrary prime ring it is proved that if $R$ admits a left $(\theta ,\phi )$-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of $R$, then $d=0$ on $R$.},
author = {Ashraf, Mohammad},
journal = {Archivum Mathematicum},
keywords = {Lie ideals; prime rings; derivations; Jordan left derivations; left derivations; torsion free rings; Lie ideals; Jordan left derivations},
language = {eng},
number = {2},
pages = {157-166},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On left $(\theta ,\varphi )$-derivations of prime rings},
url = {http://eudml.org/doc/249503},
volume = {041},
year = {2005},
}

TY - JOUR
AU - Ashraf, Mohammad
TI - On left $(\theta ,\varphi )$-derivations of prime rings
JO - Archivum Mathematicum
PY - 2005
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 041
IS - 2
SP - 157
EP - 166
AB - Let $R$ be a $2$-torsion free prime ring. Suppose that $\theta , \phi $ are automorphisms of $R$. In the present paper it is established that if $R$ admits a nonzero Jordan left $(\theta ,\theta )$-derivation, then $R$ is commutative. Further, as an application of this resul it is shown that every Jordan left $(\theta ,\theta )$-derivation on $R$ is a left $(\theta ,\theta )$-derivation on $R$. Finally, in case of an arbitrary prime ring it is proved that if $R$ admits a left $(\theta ,\phi )$-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal of $R$, then $d=0$ on $R$.
LA - eng
KW - Lie ideals; prime rings; derivations; Jordan left derivations; left derivations; torsion free rings; Lie ideals; Jordan left derivations
UR - http://eudml.org/doc/249503
ER -

References

top
  1. Ashraf M., Rehman N., On Lie ideals and Jordan left derivation of prime rings, Arch. Math.(Brno) 36 (2000), 201–206. MR1785037
  2. Ashraf M., Rehman N., Quadri M. A., On ( σ , τ ) - derivations in certain class of rings, Rad. Mat. 9 (1999), 187–192. (1999) MR1790206
  3. Ashraf M., Rehman N., Quadri M. A., On Lie ideals and ( σ , τ ) -Jordan derivations on prime rings, Tamkang J. Math. 32 (2001), 247–252. Zbl1006.16044MR1865617
  4. Ashraf M., Rehman N., Shakir Ali, On Jordan left derivations of Lie ideals in prime rings, Southeast Asian Bull. Math. 25 (2001), 375–382. 
  5. Awtar R., Lie and Jordan structures in prime rings with derivations, Proc. Amer. Math. Soc. 41 (1973), 67–74. (1973) MR0318233
  6. Awtar R., Lie ideals and Jordan derivations of prime rings, Proc. Amer. Math. Soc. 90 (1984), 9–14. (1984) Zbl0528.16020MR0722405
  7. Bell H. E., Daif M. N., On derivations and commutativity in prime rings, Acta Math. Hungar. 66 (1995), 337–343. (1995) Zbl0822.16033MR1314011
  8. Bell H. E., Kappe L. C., Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53 (1989), 339–346. (1989) Zbl0705.16021MR1014917
  9. Bergen J., Herstein I. N., Kerr J. W., Lie ideals and derivations of prime rings , J. Algebra 71 (1981), 259–267. (1981) Zbl0463.16023MR0627439
  10. Bresar M., Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003–1006. (1988) Zbl0691.16039MR0929422
  11. Bresar M., Vukman J., Jordan ( θ , φ ) -derivations, Glas. Mat., III. Ser. 26 (1991), 13–17. (1991) MR1269170
  12. Bresar M., Vukman J., On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), 7–16. (1990) Zbl0703.16020MR1028284
  13. Cairini L., Giambruno A., Lie ideals and nil derivations, Boll. Un. Mat. Ital. 6 (1985), 497–503. (1985) MR0821089
  14. Deng Q., Jordan ( θ , φ ) -derivations, Glasnik Mat. 26 (1991), 13–17. (1991) MR1269170
  15. Herstein I. N., Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110. (1957) MR0095864
  16. Herstein I. N., Topics in ring theory, Univ. of Chicago Press, Chicago 1969. (1969) Zbl0232.16001MR0271135
  17. Kill-Wong Jun, Byung-Do Kim, A note on Jordan left derivations, Bull. Korean Math. Soc. 33 (1996), 221–228. (1996) MR1405475
  18. Posner E. C., Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. (1957) MR0095863
  19. Vukman J.,, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), 47–52. (1990) Zbl0697.16035MR1007517
  20. Vukman J., Jordan left derivations on semiprime rings, Math. J. Okayama Univ. 39 (1997), 1–6. (1997) Zbl0937.16044MR1680747

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.