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Jordan normal form projections. (English) Zbl 0636.15005

The following is the main theorem in the paper. Let R be a commutative ring with 1, with polynomial extension ring R[t]. Let T:V\(\to V\) be an endomorphism of an R-module V such that there exists a completely reducible annihilating polynomial of degree n \[ \phi (t)=(t-\lambda_ 1)^{n_ 1}(t-\lambda_ 2)^{n_ 2}...(t-\lambda_ m)^{n_ m}\in R[t]\quad (n=\sum^{m}_{j=1}n_ j), \] with the eigenvalues \(\lambda_ 1,\lambda_ 2,...,\lambda_ m\in R\) such that each \(\lambda_ j-\lambda_ k\in R\) (j\(\neq k)\) is a unit. For \(j=1,2,...,m\) define the polynomial of degree \((n-n_ j)\) \(g_ j(t)=\prod_{k\neq j}(t-\lambda_ k)^{n_ k}\in R[t],\)
so that \(\phi (t)=(t-\lambda_ j)^{n_ j}g_ j(t).\) Then \(g_ j(\lambda_ j)^{-1}g_ j(T)V\to V\) is a near-projection in the endomorphism ring of V, and its associated projection \(p_ j(T)=(g_ j(\lambda_ j)^{-1}g_ j(T))_{\omega}:V\to V\) is the projection onto the direct summand \[ V_ j=im(g_ j(T):V\to V)=\ker ((T- \lambda_ jI)^{n_ j}:V\to V)\subseteq V, \] with \(\sum^{m}_{j=1}p_ j(T)=1:V\to V\), \(p_ j(T)p_ k(T)=0\) for \(j\neq k\). V is the direct sum of the T-invariant submodules \(V_ j(j=1,2,...,m)\), such that \(T-\lambda_ jI:V_ k\to V_ k\) is nilpotent for \(j=k\) and an automorphism for \(j\neq k\).
Reviewer: Yueh-er Kuo

MSC:

15A21 Canonical forms, reductions, classification
15A54 Matrices over function rings in one or more variables
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References:

[1] G. Almkvist, Endomorphisms of finitely generated projective modules over a commutative ring. Ark. Math.11, 263-301 (1974). · Zbl 0278.13005 · doi:10.1007/BF02388522
[2] F. R.Gantmacher, Matrizenrechnung I. Allgemeine Theorie. Berlin 1958.
[3] W.Lück and A.Ranicki, Chain homotopy projections. Preprint 1986. · Zbl 0616.57011
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