Derivations with Engel conditions in prime and semiprime rings

Shuliang Huang

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 4, page 1135-1140
  • ISSN: 0011-4642

Abstract

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Let R be a prime ring, I a nonzero ideal of R , d a derivation of R and m , n fixed positive integers. (i) If ( d [ x , y ] ) m = [ x , y ] n for all x , y I , then R is commutative. (ii) If Char R 2 and [ d ( x ) , d ( y ) ] m = [ x , y ] n for all x , y I , then R is commutative. Moreover, we also examine the case when R is a semiprime ring.

How to cite

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Huang, Shuliang. "Derivations with Engel conditions in prime and semiprime rings." Czechoslovak Mathematical Journal 61.4 (2011): 1135-1140. <http://eudml.org/doc/196816>.

@article{Huang2011,
abstract = {Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m, n$ fixed positive integers. (i) If $(d[x,y])^\{m\}=[x,y]_\{n\}$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathop \{\rm Char\}R\ne 2$ and $[d(x),d(y)]_\{m\}=[x,y]^\{n\}$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.},
author = {Huang, Shuliang},
journal = {Czechoslovak Mathematical Journal},
keywords = {prime and semiprime rings; ideal; derivation; GPIs; prime rings; semiprime rings; derivations; GPIs; Engel conditions; commutativity theorems},
language = {eng},
number = {4},
pages = {1135-1140},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Derivations with Engel conditions in prime and semiprime rings},
url = {http://eudml.org/doc/196816},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Huang, Shuliang
TI - Derivations with Engel conditions in prime and semiprime rings
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 1135
EP - 1140
AB - Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m, n$ fixed positive integers. (i) If $(d[x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathop {\rm Char}R\ne 2$ and $[d(x),d(y)]_{m}=[x,y]^{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.
LA - eng
KW - prime and semiprime rings; ideal; derivation; GPIs; prime rings; semiprime rings; derivations; GPIs; Engel conditions; commutativity theorems
UR - http://eudml.org/doc/196816
ER -

References

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  1. Beidar, K. I., III, W. S. Martindale, Mikhalev, A. V., Rings with Generalized Identities, Pure and Applied Mathematics, Marcel Dekker 196, New York (1996). (1996) MR1368853
  2. Bell, H. E., Daif, M. N., 10.1007/BF01876049, Acta Math. Hung. 66 (1995), 337-343. (1995) Zbl0822.16033MR1314011DOI10.1007/BF01876049
  3. Bell, H. E., Daif, M. N., 10.4153/CMB-1994-064-x, Can. Math. Bull. 37 (1994), 443-447. (1994) Zbl0820.16031MR1303669DOI10.4153/CMB-1994-064-x
  4. Chuang, Ch.-L., 10.1090/S0002-9939-1988-0947646-4, Proc. Am. Math. Soc. 103 (1988), 723-728. (1988) Zbl0656.16006MR0947646DOI10.1090/S0002-9939-1988-0947646-4
  5. Chuang, Ch.-L., 10.1006/jabr.1994.1140, J. Algebra 166 (1994), 34-71. (1994) Zbl0805.16035MR1276816DOI10.1006/jabr.1994.1140
  6. Lanski, Ch., 10.1090/S0002-9939-1993-1132851-9, Proc. Am. Math. Soc. 118 (1993), 731-734. (1993) Zbl0821.16037MR1132851DOI10.1090/S0002-9939-1993-1132851-9
  7. Daif, M. N., Bell, H. E., 10.1155/S0161171292000255, Int. J. Math. Math. Sci. 15 (1992), 205-206. (1992) Zbl0746.16029MR1143947DOI10.1155/S0161171292000255
  8. Dhara, B., Sharma, R. K., 10.1007/s11202-009-0007-6, Sib. Math. J. 50 (2009), 60-65. (2009) Zbl1205.16032MR2502875DOI10.1007/s11202-009-0007-6
  9. Erickson, T. S., III, W. S. Martindale, Osborn, J. M., 10.2140/pjm.1975.60.49, Pac. J. Math. 60 (1975), 49-63. (1975) MR0382379DOI10.2140/pjm.1975.60.49
  10. Herstein, I. N., 10.1016/0021-8693(79)90102-9, J. Algebra 60 (1979), 567-574. (1979) Zbl0436.16014MR0549949DOI10.1016/0021-8693(79)90102-9
  11. Jacobson, N., Structure of Rings, Colloquium Publications 37, Am. Math. Soc. VII, Provindence, RI (1956). (1956) Zbl0073.02002MR0081264
  12. Kharchenko, V. K., 10.1007/BF01670115, Algebra Logic 17 (1979), 155-168. (1979) MR0541758DOI10.1007/BF01670115
  13. Lin, J.-S., Liu, Ch.-K., Strong commutativity preserving maps on Lie ideals, Linear Algebra Appl. 428 (2008), 1601-1609. (2008) Zbl1141.16021MR2388643
  14. Lee, T.-K., Semiprime rings with differential identities, Bull. Inst. Math., Acad. Sin. 20 (1992), 27-38. (1992) Zbl0769.16017MR1166215
  15. Mayne, J. H., 10.4153/CMB-1984-018-2, Can. Math. Bull. 27 (1984), 122-126. (1984) Zbl0537.16029MR0725261DOI10.4153/CMB-1984-018-2
  16. III, W. S. Martindale, 10.1016/0021-8693(69)90029-5, J. Algebra 12 (1969), 576-584. (1969) MR0238897DOI10.1016/0021-8693(69)90029-5
  17. Posner, E. C., 10.1090/S0002-9939-1957-0095863-0, Proc. Am. Math. Soc. 8 (1958), 1093-1100. (1958) Zbl0082.03003MR0095863DOI10.1090/S0002-9939-1957-0095863-0

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