Block diagonalization and $2$-unit sums of matrices over Prüfer domains
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- by Peter Vámos and Sylvia Wiegand PDF
- Trans. Amer. Math. Soc. 363 (2011), 4997-5020 Request permission
Abstract:
We show that matrices over a large class of Prüfer domains are equivalent to “almost diagonal” matrices, that is, to matrices with all the nonzero entries congregated in blocks along the diagonal, where both dimensions of the diagonal blocks are bounded by the size of the class group of the Prüfer domain. This result, a generalization of a 1972 result of L. S. Levy for Dedekind domains, implies that, for $n$ sufficiently large, every $n\times n$ matrix is a sum of two invertible matrices. We also generalize from Dedekind to certain Prüfer domains a number of results concerning the presentation of modules and the equivalence of matrices presenting them, and we uncover some connections to combinatorics.References
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Additional Information
- Peter Vámos
- Affiliation: Department of Mathematical Sciences, University of Exeter, Exeter EX4 4QF, England
- Email: P.V{\' a}mos@exeter.ac.uk
- Sylvia Wiegand
- Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
- MR Author ID: 182675
- Email: swiegand@math.unl.edu
- Received by editor(s): September 18, 2009
- Received by editor(s) in revised form: October 21, 2009, and February 18, 2010
- Published electronically: April 14, 2011
- Additional Notes: The authors thank the Universities of Exeter and Nebraska for their hospitality and the University of Nebraska Research Council and the London Mathematical Society for providing funds so that the first author could spend about one month at Nebraska in the fall of 2005, when this research was begun. We thank Roger Wiegand and Lee Klingler for their helpful suggestions.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 4997-5020
- MSC (2010): Primary 13F05, 13F07, 13G05, 13A15, 13C05, 13C20, 15A21, 16U60
- DOI: https://doi.org/10.1090/S0002-9947-2011-05328-6
- MathSciNet review: 2806699