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On Rings With Engel Cycles

Published online by Cambridge University Press:  20 November 2018

H. E. Bell
Affiliation:
Mathematics Department, Brock University, St. Catharines, Ontario L2S 3A1
A. A. Klein
Affiliation:
Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel-Aviv, Israel
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Abstract

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A ring R is called an EC-ring if for each x, y ∊ R, there exist distinct positive integers m, n such that the extended commutators [x, y]m and [x, y]n are equal. We show that in certain EC-rings, the commutator ideal is periodic; we establish commutativity of arbitrary EC-domains; we prove that a ring R is commutative if for each x, y ∊ R, there exists n > 1 for which [x, y] = [x, y]n.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bell, H. E., Some commutativity results for periodic rings , Acta Math. Acad. Sci. Hungar. 28(1976), 279 283.Google Scholar
2. Bell, H. E., On commutativity of periodic rings and near-rings, Acta Math. Acad. Sci. Hungar. 36 (1980), 293302.Google Scholar
3. Bell, H. E. and Kappe, L. C., Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53 (1989), 339346.Google Scholar
4. Bergen, J. and Herstein, I. N., The algebraic hypercenter and some applications , J. Algebra 85(1983), 217 242.Google Scholar
5. Brandi, R., Infinite soluble groups with Engel cycles; afiniteness condition , Math. Z. 182(1983), 259264.Google Scholar
6. Chacron, M., On a theorem of Herstein, Canad. J. Math. 21 (1969), 13481353.Google Scholar
7. Herstein, I. N., Sugli anelli soddisfacenti ad una condizione di Engel, Atti. Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Nat. (8)32 (1962), 177180.Google Scholar
8. Herstein, I. N., A remark on rings and algebras, Michigan Math. J. 10 (1963), 269272.Google Scholar
9. Streb, W., Uber einen Satz von Herstein undNakayama, Red. Sem. Mat. Univ. Padova 64 (1981), 159171.Google Scholar