## 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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.