Generalized derivations in prime and semiprime
DOI:
https://doi.org/10.5269/bspm.v34i2.21774Resumo
Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$ fixed positive integers. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for all $x,y\in I$, then $R$ is commutative. Moreover we also examine the case when $R$ is a semiprime ring.
Referências
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2. N. Argac and H. G. Inceboz, Derivations of prime and semiprime rings, J. Korean Math. Soc., 46(2009), no.5, 997-1005.
3. K. I. Beidar, W. S. Martindale and V. Mikhalev, Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, 196. Marcel Dekker, Inc., New York, 1996.
4. M. Bresar, On the distance of the composition of two derivations to be the generalized derivations, Glasgow Math. J., 33(1991), 89-93.
5. C. L. Chuang, GPIs having coefficents in Utumi quotient rings, Proc. Amer. Math. Soc., 103(1988), no.3, 723-728.
6. C. L. Chuang, Hypercentral derivations, J. Algebra, 161(1994), 37-71.
7. M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Internt. J. Math. & Math. Sci., 15(1992), 205-206.
8. J. S. Erickson, W. S. Martindale III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60(1975), no.1, 49-63.
9. B. Hvala, Generalized derivations in prime rings, Comm. Algebra, 26(1998), no.4, 1147-1166.
10. S. L. Huang, On generalized derivations of prime and semiprime rings, Taiwanese Journal of Mathematics (to appear).
11. I. N. Herstein, Center-like elements in prime rings, J. Algebra, 60(1979), 567-574.
12. V. K. Kharchenko, Differential identities of prime rings, Algebra and Logic, 17(1978), 155-168.
13. T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra, 27(1998), no.8, 4057-4073.
14. C. Lanski, An engel condition with derivation, Proc. Amer. Math. Soc., 118(1993), no.3, 731-734.
15. T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20(1)(1992), 27-38
16. W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12(1969), 176-584.
17. J. H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull., 27(1984), no.1, 122-126.
18. M. A. Quadri, M. S. Khan and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J.pure appl. Math., 34(2003), no.98, 1393-1396.
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2015-05-06
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