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Prime rings with hypercommuting derivations on a Lie ideal. (English) Zbl 0943.16015

The authors prove a result involving generalized commutators with derivation. Let \(R\) be a prime ring, \(D\) a nonzero derivation of \(R\), and \(L\) a noncentral Lie ideal of \(R\). Set \([a,b]_1=ab-ba\) and for \(k>1\) \([a,b]_k=[[a,b]_{k-1},b]\). If for each \(u\in L\) \([D(u^m),u^m]_n=0\), where \(m=m(u)\geq 1\) and \(n=n(u)\geq 1\), then when \(R\) contains no nonzero nil right ideal, \(R\) must satisfy the standard polynomial identity \(S_4\).

MSC:

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16U80 Generalizations of commutativity (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
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References:

[1] K.I. Beidar - W.S. Martindale III - V. Mikhalev , Rings with generalized identities , Pure and Applied Math. , Dekker , New York ( 1996 ). MR 1368853 | Zbl 0847.16001 · Zbl 0847.16001
[2] C.L. Chuang , GPIs having coefficients in Utumi quotient ring , Proc. Amer. Math. Soc. 103 , no. 3 ( 1988 ). MR 947646 | Zbl 0656.16006 · Zbl 0656.16006
[3] C.L. Chuang , Hypercentral derivations , J. Algebra , 166 ( 1994 ), pp. 34 - 71 . MR 1276816 | Zbl 0805.16035 · Zbl 0805.16035
[4] C.L. Chuang - J.S. Lin , On a conjecture by Herstein , J. Algebra , 126 ( 1989 ), pp. 119 - 138 . MR 1023288 | Zbl 0688.16036 · Zbl 0688.16036
[5] V. De FILIPPIS - O. M. DI VINCENZO, On the generalized hypercentralizer of a Lie ideal in a prime ring , Rend. Sem. Mat. Univ. Padova , 100 ( 1998 ), pp. 283 - 295 . Numdam | MR 1675291 | Zbl 0921.16011 · Zbl 0921.16011
[6] O.M. Di Vincenzo , On the n-th centralizer of a Lie ideal , Boll. UMI ( 7 ), 3-A ( 1989 ), pp. 77 - 85 . MR 990089 | Zbl 0692.16022 · Zbl 0692.16022
[7] O.M. Di Vincenzo , Derivations and multilinear polynomials , Rend. Sem. Mat. Univ. Padova , 81 ( 1989 ), pp. 209 - 219 . Numdam | MR 1020195 | Zbl 0738.16016 · Zbl 0738.16016
[8] C. Faith , Lectures on Injective Modules and Quotient Rings , Lecture Notes in Mathematics , 49 , Springer-Verlag , New York ( 1967 ). MR 227206 | Zbl 0162.05002 · Zbl 0162.05002
[9] I.N. Herstein , Rings with involution , Univ. of Chicago Press , Chicago ( 1976 ). MR 442017 | Zbl 0343.16011 · Zbl 0343.16011
[10] I.N. Herstein , On the hypercenter of a ring , J. Algebra , 36 ( 1975 ), pp. 151 - 157 . MR 371962 | Zbl 0313.16036 · Zbl 0313.16036
[11] I.N. Herstein , A theorem on invariant subrings , J. Algebra , 83 ( 1983 ), pp. 26 - 32 . MR 710584 | Zbl 0514.16001 · Zbl 0514.16001
[12] N. Jacobson - P.I. Algebras , An Introduction , Lecture Notes in Mathematics , no. 44 , Springer-Verlag , Berlin / New York ( 1975 ). MR 369421 | Zbl 0326.16013 · Zbl 0326.16013
[13] V.K. Kharchenko , Differential identities of prime rings , Algebra and Logic , 17 ( 1978 ), pp. 155 - 168 . MR 541758 | Zbl 0423.16011 · Zbl 0423.16011
[14] V.K. Kharchenko , Differential identities of semiprime rings , Algebra and Logic , 18 ( 1979 ), pp. 86 - 119 . MR 566776 | Zbl 0464.16027 · Zbl 0464.16027
[15] J. Lambek , Lectures on Rings and Modules , Blaisdell Waltham, MA ( 1966 ). MR 206032 | Zbl 0143.26403 · Zbl 0143.26403
[16] T.K. Lee , Semiprime rings with differential identities , Bull. Inst. Math. Acad. Sinica , 20 , no. 1 ( 1992 ), pp. 27 - 38 . MR 1166215 | Zbl 0769.16017 · Zbl 0769.16017
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