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The Fredholm Elements of a Ring

Published online by Cambridge University Press:  20 November 2018

Bruce Alan Barnes
Bruce Alan Barnes*
Affiliation:
The University of Oregon, Eugene, Oregon
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In (1), Atkinson characterized the set of Fredholm operators on a Banach space X as those bounded operators invertible modulo the two-sided ideal of compact operators on X. It follows from this characterization that the Fredholm operators can also be described as those bounded operators which are invertible modulo the two-sided ideal of bounded operators on X which have finite-dimensional range. This ideal is the socle of the algebra of all bounded operators on X. Now, if A is any ring with no nilpotent left or right ideals, then the concept of socle makes sense (the socle of A in this case is the algebraic sum of the minimal left ideals of A, or 0 if A has no minimal left ideals). Also, in this case, the socle is a two-sided ideal of A. In this paper we study the elements in a ring A which are invertible modulo the socle. We call these elements the Fredholm elements of A.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 84 - 95
Copyright
Copyright © Canadian Mathematical Society 1969

References

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