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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Block diagonalization and $ 2$-unit sums of matrices over Prüfer domains


Authors: Peter Vámos and Sylvia Wiegand
Journal: Trans. Amer. Math. Soc. 363 (2011), 4997-5020
MSC (2010): Primary 13F05, 13F07, 13G05, 13A15, 13C05, 13C20, 15A21, 16U60
Published electronically: April 14, 2011
MathSciNet review: 2806699
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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.


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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
Email: swiegand@math.unl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05328-6
Keywords: Matrices, sums of units, Prüfer domains, monoids, Picard group, class group
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.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.