Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T02:12:57.661Z Has data issue: false hasContentIssue false
  • English
  • Français

N-th root rings

Published online by Cambridge University Press:  17 April 2009

Henry Heatherly  and
Altha Blanchet
Henry Heatherly
Affiliation:
Department of Mathematics, University of Southwestern Louisiana, Lafayette, Louisana 70504, United Stats of America.
Altha Blanchet
Affiliation:
Department of Mathematics, University of Southwestern Louisiana, Lafayette, Louisana 70504, United Stats of America.
Rights & Permissions [Opens in a new window]

Abstract

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.

A ring for which there is a fixed integer n ≥ 2 such that every element in the ring has an n-th in the ring is called an n-th root ring. This paper gives numerous examples of diverse types of n-th root rings, some via general construction procedures. It is shown that every commutative ring can be embedded in a commutative n-th root ring with unity. The structure of n-th root rings with chain conditions is developed and finite n-th root rings are completely classified. Subdirect product representations are given for several classes of n-th root rings.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 35 , Issue 1 , February 1987 , pp. 111 - 123
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Abian, A., “Conditionally complete and conditionally orthogonally complete rings”, Czechoslovak. Math. J. 30 (1980), 171–176.CrossRefGoogle Scholar
[2]Andrunakievic, V. and Rjabuhin, Ju. M., “Rings without nilpotent elements and completely simple ideals”, Soviet Math. Dokl. 9 (1968) 565–567.Google Scholar
[3]Divinsky, N., “Commutative subdirectly irreducible rings”, Proc. Amer. Math. Soc. 8 (1957), 642–648.CrossRefGoogle Scholar
[4]Klein, A., “A simple proof of a theorem on reduced rings”, Canad. Math. Bull. 23 (1980), 495–496.CrossRefGoogle Scholar
[5]Koh, K., “On very large one sided ideals of a ring”, Canad. Math. Bull. 9 (1960), 191–196.CrossRefGoogle Scholar
[6]McCoy, N., “Subdirectly irreducible commutative rings”, Duke Math. J. 12 (1945), 381–387.CrossRefGoogle Scholar
[7]Mikusinski, J., Operation Calculus, Pure and Applied Mathematics 8 (Pergamon, Press, Oxford, 1959).Google Scholar
[8]Niven, I., “Equations in quaternions”, Amer. Math. Monthly 48 (1941), 654–661.CrossRefGoogle Scholar
[9]Redei, L., Algebra, Pure and Applied Mathematics 91 (Pergamon Press, Oxford, 1967).Google Scholar
[10]Stenström, B., Rings of Quotients, Grundlehren der mathematischen Wissenschaften 217 (Springer-Verlag, New York, 1975).CrossRefGoogle Scholar