Results in Mathematics

, Volume 42, Issue 1–2, pp 3–8

# On Commutativity of Rings With Derivations

Article

## Abstract

Let R be a ring and d : R → R a derivation of R. In the present paper we investigate commutativity of R satisfying any one of the properties (i)d([x,y]) = [x,y], (ii)d(x o y) = xoy, (iii)d(x) o d(y) = 0, or (iv)d(x) o d(y) = x o y, for all x, y in some apropriate subset of R.

## Keywords and Phrases

prime rings derivations ideals Lie ideals and commutativity

## 1991 Mathematics Subject Classification

16W25 16N60 16U80

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## References

1. [1]
Awtar, R., Lie structure in prime rings with derivations, Publ. Math.(Debrecen) 31 (1984), 209–215.
2. [2]
Awtar, R., Lie and Jordan structures in prime rings with derivations, Proc. Amer. Math. Soc. 41 (1973), 67–74.
3. [3]
Bell, H.E. and Daif, M.N., On commutativity and strong commutativity preserving maps, Canad. Math. Bull. 37 (1994), 443–447.
4. [4]
Bell, H. E. and Martindale, W. S., Centralizing mappings of semiprime rings, Canad.Math. Bull. 30 (1987), 92–101.
5. [5]
Bergen, J., Herstein, I. N. and Kerr, J. W., Lie ideals and derivations of prime rings, J. Algebra 71 (1981), 259–267.
6. [6]
Bresar, M., Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings, Trans.Amer. Math. Soc. 335 (1993), 525–546.
7. [7]
Bresar, M., Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385–394.
8. [8]
Daif, M. N. and Bell, H. E., Remarks on derivations on semiprime rings, Internal. J. Math, &: Math. Sci. 15 (1992), 205–206.
9. [9]
Deng, Q. and Ashraf, M., On strong commutativity preserving mappings, Results in Math. 30 (1996), 259–263.
10. [10]
Herstein, I. N., Ring with involution, Univ. Chicago press, Chicago 1976.Google Scholar
11. [11]
Herstein, I. N., Topics in ring theory, Univ. Chicago press, Chicago 1969.
12. [12]
Hongan, M., A note on semiprime rings with derivations, Internat. J. Math. & Math. Sci. 20 (1997), 413–415.
13. [13]
Mayne, J.H., Centralizing mappings of prime rings, Canad. Math. Bull. 27 (1984), 122–126.
14. [14]
Posner, E. C, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.