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

Vámos, Peter; Wiegand, Sylvia

doi:10.1090/S0002-9947-2011-05328-6

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.

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