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Are One-Sided Inverses Two-Sided Inverses in a Matrix Ring Over a Group Ring?

Published online by Cambridge University Press:  20 November 2018

Gerald Losey
Gerald Losey*
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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A ring R with identity element is n-finite if for any pair A, B of n × n matrices over R, AB = In implies BA = In. In module theoretic terms, R is n-finite if and only if in a free R-module of rank n any generating set of n elements is free. If R is n-finite for all positive integers n then R is said to be strongly finite. It is known that all commutative rings, all Artinian rings and all Noetherian rings are strongly finite. These and many other interesting results appear in a paper of P. M. Cohn [1].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 4 , December 1970 , pp. 475 - 479
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Cohn, P. M., Some remarks on the invariant basis property, Topology, 5 (1966), 215-228.Google Scholar
2. Jennings, S. A., The group ring of a class of infinite nilpotent groups, Canad. J. Math. 7 (1955), 169-187.Google Scholar
3. Kaplansky, I., Rings of operators, Benjamin, New York, 1969.Google Scholar
4. Montgomery, M. S., Left and ring inverses in group algebras, Bull. Amer. Math.Soc, 75 (1969), 539-540.Google Scholar
5. Neumann, H., Varieties of Groups, Springer-Verlag, New York, 1967.Google Scholar
6. Zariski, O. and Samuel, P., Commutative Algebra, I, Van Nostrand, Princeton, 1958.Google Scholar