| Title:
|
On $(\sigma,\tau)$-derivations in prime rings (English) |
| Author:
|
Ashraf, Mohammad |
| Author:
|
Nadeem-ur-Rehman |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
38 |
| Issue:
|
4 |
| Year:
|
2002 |
| Pages:
|
259-264 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $R$ be a 2-torsion free prime ring and let $\sigma , \tau $ be automorphisms of $R$. For any $x, y \in R$, set $[x , y]_{\sigma , \tau } = x\sigma (y) - \tau (y)x$. Suppose that $d$ is a $(\sigma , \tau )$-derivation defined on $R$. In the present paper it is shown that $(i)$ if $R$ satisfies $[d(x) , x]_{\sigma , \tau } = 0$, then either $d = 0$ or $R$ is commutative $(ii)$ if $I$ is a nonzero ideal of $R$ such that $[d(x) , d(y)] = 0$, for all $x, y \in I$, and $d$ commutes with both $\sigma $ and $\tau $, then either $d = 0$ or $R$ is commutative. $(iii)$ if $I$ is a nonzero ideal of $R$ such that $d(xy) = d(yx)$, for all $x, y \in I$, and $d$ commutes with $\tau $, then $R$ is commutative. Finally a related result has been obtain for $(\sigma , \tau )$-derivation. (English) |
| Keyword:
|
prime rings |
| Keyword:
|
$(\sigma, \tau )$-derivations |
| Keyword:
|
torsion free rings and commutativity |
| MSC:
|
16N60 |
| MSC:
|
16U70 |
| MSC:
|
16U80 |
| MSC:
|
16W25 |
| idZBL:
|
Zbl 1068.16047 |
| idMR:
|
MR1942655 |
| . |
| Date available:
|
2008-06-06T22:30:51Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107839 |
| . |
| Reference:
|
[1] Aydin N., Kaya A.: Some generalization in prime rings with $(\sigma , \tau )$-derivation.Doga Turk. J. Math. 16 (1992), 169–176. MR 1202970 |
| Reference:
|
[2] Bell H. E., Martindale W. S.: Centralizing mappings of semiprime rings.Canad. Math. Bull. 30 (1987), 92–101. Zbl 0614.16026, MR 0879877 |
| Reference:
|
[3] Bell H. E., Kappe L. C.: Ring in which derivations satisfy certain algebric conditions.Acta Math. Hungar. 53 (1989), 339–346. MR 1014917 |
| Reference:
|
[4] Bell H. E., Daif M. N.: On commutativity and strong commutativity preserving maps.Canad. Math. Bull. 37 (1994), 443–447. Zbl 0820.16031, MR 1303669 |
| Reference:
|
[5] Bell H. E., Daif M. N.: On derivations and commutativity in prime rings.Acta Math. Hungar. 66 (1995), 337–343. Zbl 0822.16033, MR 1314011 |
| Reference:
|
[6] Bresar M.: On a generalization of the notion of centralizing mappings.Proc. Amer. Math. Soc. 114 (1992), 641–649. Zbl 0754.16020, MR 1072330 |
| Reference:
|
[7] Bresar M.: Centralizing mappings and derivations in prime rings.J. Algebra 156 (1993), 385–394. Zbl 0773.16017, MR 1216475 |
| Reference:
|
[8] Daif M. N., Bell H. E.: Remarks on derivations on semiprime rings.Int. J. Math. Math. Sci. 15 (1992), 205–206. Zbl 0746.16029, MR 1143947 |
| Reference:
|
[9] Herstein I. N.: A note on derivations.Canad. Math. Bull. 21 (1978), 369–370. Zbl 0412.16018, MR 0506447 |
| Reference:
|
[10] Herstein I. N.: Rings with involution.Univ. Chicago Press, Chicago 1976. Zbl 0343.16011, MR 0442017 |
| Reference:
|
[11] Posner E. C.: Derivations in prime rings.Proc. Amer. Math. Soc. 8 (1957), 1093–1100. MR 0095863 |
| Reference:
|
[12] Vukman J.: Commuting and centralizing mappings in prime rings.Proc. Amer. Math. Soc. 109 (1990), 47–52. Zbl 0697.16035, MR 1007517 |
| Reference:
|
[13] Vukman J.: Derivations on semiprime rings.Bull. Austral. Math. Soc. 53 (1995), 353–359. MR 1388583 |
| . |
