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One-Sided Inverses in Rings

Published online by Cambridge University Press:  20 November 2018

Roger D. Peterson
Roger D. Peterson*
Affiliation:
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin
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Following Herstein [2], we will call a ring R with identity von Neumann finite (vNf) provided that xy = 1 implies yx — WnR. Kaplansky [4] showed that group algebras over fields of characteristic zero are vNf rings, and further, that full matrix rings over such rings are also vNf. Herstein [2] has posed the problem for group algebras over fields of arbitrary characteristic. If group algebras over fields are always vNf, then it is easily seen that group algebras over commutative rings are always vNf. What conditions on the underlying ring of scalars would force the vNf property for all group rings over it?

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 218 - 224
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Baer, R., Inverses and zero divisors, Bull. Amer. Math. Soc. 48 (1942), 630–638.Google Scholar
2. Herstein, I. N., Notes from a ring theory conference, Regional Conference Series in Mathematics No. 9 (Amer. Math. Soc, Providence, 1971).Google Scholar
3. Jacobson, N., Some remarks on one-sided inverses, Proc. Amer. Math. Soc. 1 (1950), 352–355.Google Scholar
4. Kaplansky, I., Fields and rings, Chicago Lectures in Mathematics (Univ. Chicago Press, Chicago, 1969).Google Scholar
5. Montgomery, M. S., Left and right inverses in group algebras, Bull. Amer. Math. Soc. 75 (1969), 539–540.Google Scholar
6. Passman, D. S., Infinite group rings (Marcel Dekker, New York, 1971).Google Scholar
7. Shepherdson, J. C., Inverses and zero divisors in matrix rings, Proc. London Math. Soc. 1 (1951), 71–85.Google Scholar