Free actions on semiprime rings

Muhammad Anwar Chaudhry; Mohammad S. Samman

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 2, page 197-208
  • ISSN: 0862-7959

Abstract

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We identify some situations where mappings related to left centralizers, derivations and generalized ( α , β ) -derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation T , of a semiprime ring R the mapping ψ R → R defined by ψ ( x ) = T ( x ) x - x T ( x ) for all x ∈ R is a free action. We also show that for a generalized ( α , β ) -derivation F of a semiprime ring R , with associated ( α , β ) -derivation d , a dependent element a of F is also a dependent element of α + d . Furthermore, we prove that for a centralizer f and a derivation d of a semiprime ring R , ψ = d ∘ f is a free action.

How to cite

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Chaudhry, Muhammad Anwar, and Samman, Mohammad S.. "Free actions on semiprime rings." Mathematica Bohemica 133.2 (2008): 197-208. <http://eudml.org/doc/250516>.

@article{Chaudhry2008,
abstract = {We identify some situations where mappings related to left centralizers, derivations and generalized $(\alpha ,\beta )$-derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation $T$, of a semiprime ring $R$ the mapping $\psi \: R \rightarrow R$ defined by $\psi (x)=T(x) x - x T(x)$ for all $x \in R$ is a free action. We also show that for a generalized $(\alpha , \beta )$-derivation $F$ of a semiprime ring $R,$ with associated $(\alpha , \beta )$-derivation $d,$ a dependent element $a$ of $F$ is also a dependent element of $\alpha + d.$ Furthermore, we prove that for a centralizer $f$ and a derivation $d$ of a semiprime ring $R$, $\psi = d\circ f$ is a free action.},
author = {Chaudhry, Muhammad Anwar, Samman, Mohammad S.},
journal = {Mathematica Bohemica},
keywords = {prime ring; semiprime ring; dependent element; free action; centralizer; derivation; prime rings; semiprime rings; dependent elements; free actions; left centralizers; generalized derivations},
language = {eng},
number = {2},
pages = {197-208},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Free actions on semiprime rings},
url = {http://eudml.org/doc/250516},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Chaudhry, Muhammad Anwar
AU - Samman, Mohammad S.
TI - Free actions on semiprime rings
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 2
SP - 197
EP - 208
AB - We identify some situations where mappings related to left centralizers, derivations and generalized $(\alpha ,\beta )$-derivations are free actions on semiprime rings. We show that for a left centralizer, or a derivation $T$, of a semiprime ring $R$ the mapping $\psi \: R \rightarrow R$ defined by $\psi (x)=T(x) x - x T(x)$ for all $x \in R$ is a free action. We also show that for a generalized $(\alpha , \beta )$-derivation $F$ of a semiprime ring $R,$ with associated $(\alpha , \beta )$-derivation $d,$ a dependent element $a$ of $F$ is also a dependent element of $\alpha + d.$ Furthermore, we prove that for a centralizer $f$ and a derivation $d$ of a semiprime ring $R$, $\psi = d\circ f$ is a free action.
LA - eng
KW - prime ring; semiprime ring; dependent element; free action; centralizer; derivation; prime rings; semiprime rings; dependent elements; free actions; left centralizers; generalized derivations
UR - http://eudml.org/doc/250516
ER -

References

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  1. 10.1016/0022-1236(73)90036-0, J. Funct. Anal. 13 (1973), 277–301. (1973) MR0470685DOI10.1016/0022-1236(73)90036-0
  2. 10.1093/qmath/44.2.129, Quart. J. Math. Oxford 44 (1993), 129–138. (1993) MR1222369DOI10.1093/qmath/44.2.129
  3. On the composition of ( α , β ) -derivations of rings, and application to von Neumann algebras, Acta Sci. Math. 56 (1992), 369–375. (1992) MR1222081
  4. ( α , β ) -derivations of prime rings having power central values, Bull. Inst. Math., Acad. Sin. 23 (1995), 295–303. (1995) MR1367365
  5. On ( α , β ) -derivations of semiprime rings, Demonstratio Math. 36 (2003), 283–287. (2003) MR1984339
  6. Special identities with ( α , β ) -derivations, Riv. Mat. Univ. Parma 5 (1996), 109–119. (1996) Zbl0876.16025MR1456405
  7. Dependent elements of automorphisms of a C * -algebra, Proc. Japan Acad. 48 (1972), 561–565. (1972) MR0341109
  8. 10.2996/kmj/1138846942, Kodai Math. Sem. Rep. 26 (1974), 1–21. (1974) Zbl0296.46063MR0358362DOI10.2996/kmj/1138846942
  9. Les Algebres d’Operateurs dans l’Espace Hilbertien, Gauthier-Villars, Paris, 1957. (1957) Zbl0088.32304MR0094722
  10. Rings with involution, Univ. Chicago Press, Chicago, 1976. (1976) Zbl0343.16011MR0442017
  11. 10.1080/00927879808826190, Comm. Algebra 26 (1998), 1147–1166. (1998) Zbl0899.16018MR1612208DOI10.1080/00927879808826190
  12. 10.1215/S0012-7094-69-03693-X, Duke Math. J. 36 (1969), 781–789. (1969) Zbl0184.17101MR0256181DOI10.1215/S0012-7094-69-03693-X
  13. On dependent elements in semiprime rings, Math. Japonica 47 (1998), 29–31. (1998) MR1606291
  14. 10.2307/1968693, Ann. Math. 37 (1936), 116–229. (1936) MR1503275DOI10.2307/1968693
  15. 10.2307/1968823, Ann. Math. 41 (1940), 94–161. (1940) Zbl0023.13303MR0000898DOI10.2307/1968823
  16. Dependent elements of left centralizers of semiprime rings, Preprint. MR2467193
  17. Modular Theory in Operator Algebras, Abacus Press, Kent, 1981. (1981) Zbl0504.46043MR0696172
  18. Centralizers of semiprime rings, Comment. Math. Univ. Carolin. 42 (2001), 237–245. (2001) MR1832143
  19. On dependent elements in rings, Int. J. Math. Math. Sci. 53–56 (2004), 2895–2906. (2004) MR2145367
  20. On dependent elements and related problems in rings, Int. Math. J. 6 (2005), 93–112. (2005) Zbl1158.16309MR2146333
  21. 10.1017/S000497270003820X, Bull. Austral. Math. Soc. 71 (2005), 225–234. (2005) MR2133407DOI10.1017/S000497270003820X
  22. On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32 (1991), 609–614. (1991) Zbl0746.16011MR1159807

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