A result on derivations

Authors:
Tsiu-Kwen Lee and Jer-Shyong Lin

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1687-1691

MSC (1991):
Primary 16W25

DOI:
https://doi.org/10.1090/S0002-9939-96-03234-0

MathSciNet review:
1307545

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a semiprime ring with a derivation and let be a Lie ideal of , . Suppose that for all , where is a fixed positive integer. Then for the ideal of generated by and if is 2-torsion free, then . Furthermore, is a subdirect sum of semiprime homomorphic images and with derivations and , induced canonically by , respectively such that and the image of in is commutative (central if is 2-torsion free), where denotes the image of in . Moreover, if , then . This gives Bre[??]sar's theorem without the -torsion free assumption on .

**1.**M. Bresar,*A note on derivations*, Math. J. Okayama Univ.**32**(1990), 83--88. MR**92g:16026****2.**L. Carini and A. Giambruno,*Lie ideals and nil derivations*, Boll. Un. Mat. Ital.**6**(1985), 497--503. MR**87d:16045****3.**C. L. Chuang,*GPIs having coefficients in Utumi quotient rings*, Proc. Amer. Math. Soc.**103**(1988), 723--728. MR**89e:16028****4.**J. S. Erickson, W. S. Martindale 3rd, and J. M. Osborn,*Prime nonassociative algebras*, Pacific J. Math.**60**(1975), 49--63. MR**52:3264****5.**B. Felzenszwalb and C. Lanski,*On the centralizers of ideals and nil derivations*, J. Algebra**83**(1983), 520--530. MR**84m:16028****6.**A. Giambruno and I. N. Herstein,*Derivations with nilpotent values*, Rend. Circ. Mat. Palermo**30**(1981), 199--206. MR**83g:16010****7.**I. N. Herstein,*Topics in ring theory*, Univ. of Chicago Press, Chicago, IL, 1969. MR**42:6018****8.**------,*Center-like elements in prime rings*, J. Algebra**60**(1979), 567--574. MR**80m:16006****9.**V. K. Kharchenko,*Differential identities of semiprime rings*, Algebra and Logic**18**(1979), 58--80. MR**81f:16052****10.**C. Lanski and S. Montgomery,*Lie structure of prime rings of characteristic*, Pacific J. Math.**42**(1972), 117--136. MR**48:2194****11.**C. Lanski,*Derivations with nilpotent values on Lie ideals*, Proc. Amer. Math. Soc.**108**(1990), 31--37. MR**90d:16041****12.**W. S. Martindale 3rd,*Prime rings satisfying a generalized polynomial identity*, J. Algebra**12**(1969), 576--584. MR**39:257**

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Additional Information

**Tsiu-Kwen Lee**

Affiliation:
Department of Mathematics, National Taiwan University, Taipei, Taiwan 10764, Republic of China

Email:
tklee@math.ntu.edu.tw

**Jer-Shyong Lin**

Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China

Email:
jslin@math.nthu.edu.tw

DOI:
https://doi.org/10.1090/S0002-9939-96-03234-0

Keywords:
Semiprime rings,
derivations,
Lie ideals,
GPIs,
differential identities

Received by editor(s):
March 28, 1994

Received by editor(s) in revised form:
May 9, 1994, and December 9, 1994

Communicated by:
Ken Goodearl

Article copyright:
© Copyright 1996
American Mathematical Society