Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-20T18:39:23.481Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Power Map and Ring Commutativity

Published online by Cambridge University Press:  20 November 2018

Howard E. Bell
Howard E. Bell*
Affiliation:
Department of Mathematics, Brock University, St. Catharines, Ontario, Canada, L2S 3A1
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R denote an associative ring with 1, let n be a positive integer, and let k = 1, 2, or 3. The ring R will be called an (n, k)-ring if it satisfies the identities

for all integers m with n ≤ m ≤ n + k - 1. It was shown years ago by Herstein (See [2], [9], and [10]) that for n >1, any (n, l)-ring must have nil commutator ideal C(R). Later Luh [12] proved that primary (rc, 3)-rings must in fact be commutative, and Ligh and Richoux [11] recently showed that all (n, 3)-rings are commutative.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 4 , 01 December 1978 , pp. 399 - 404
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Awtar, R., On the commutativity of non-associative rings, Publ. Math. Debrecen 22 (1975), 177-188.Google Scholar
2. Bell, H. E., On a commutativity theorem of Herstein, Arch. Math. 21 (1970), 265-267.Google Scholar
3., Bell, H. E., Certain near-rings are rings, J. London Math. Soc. (2) 4 (1971), 264-270.Google Scholar
4., Bell, H. E., On some commutativity theorems of Herstein, Arch. Math. 24 (1973), 34-38.Google Scholar
5. Harmanci, A., Two elementary commutativity theorems for rings, Acta. Math. Acad. Sci. Hungar. 29 (1977), 23-29.Google Scholar
6. Herstein, I. N., A generalization of a theorem of Jacobson, Amer. J. Math. 73 (1951), 756-762.Google Scholar
7. Herstein, I. N., A theorem on rings, Canadian J. Math. 5 (1953), 238-241.Google Scholar
8. Herstein, I. N., The structure of a certain class of rings, Amer. J. Math. 75 (1953), 864-871.Google Scholar
9. Herstein, I. N., Power maps in rings, Michigan Math. J. 8 (1961), 29-32.Google Scholar
10. Herstein, I. N., A remark on rings and algebras, Michigan Math. J. 10 (1963), 269-272 Google Scholar
11. Ligh, S. and Richoux, A., A commutativity theorem for rings, Bull. Austral. Math. Soc. 16 (1977), 75-77.Google Scholar
12. Luh, J., A commutativity theorem for primary rings, Acta. Math. Acad. Sri. Hungar. 22 (1971), 211-213.Google Scholar