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Jordan homomorphisms and harmonic mappings. (English) Zbl 1028.51004

Let \(R\) be an (associative) ring with \(1\not= 0\). The projective line consists of the points \(Ru\) where \(u\in R^2\) is an element of a basis \(u,v\) for the free \(R\)-module \(R^2\). Points \(p\) and \(q\) are called distant if one has \(p= Ru\) and \(q=Rv\) such that \(u,v\) is a basis for \(R^2\). A Jordan homomorphism \(\alpha : R\rightarrow R'\) of rings is a homomorphism for \(+\) such that \((abc)\alpha =a\alpha \cdot b\alpha \cdot c\alpha\) for all \(a,b,c\in R\). The set \(A(R)\) of points distant to the point \(R(1,0)\) consists of the points \(R(t,1)\) where \(t\in R\). A Jordan homomorphism \(\alpha : R\rightarrow R'\) yields a mapping \(\overline{\alpha } : A(R)\rightarrow A(R')\), \(R(t,1)\mapsto R'(t\alpha ,1')\).
Let \(C\) denote the connected component of \(R(1,0)\) in \({\mathbf P}(R)\) (where points have a common edge if they are distant). The authors construct an extension of \(\overline{\alpha }\) to a harmonic mapping \(C\rightarrow {\mathbf P}(R')\). Such a mapping was studied earlier when \(R\) has stable rank 2 (which implies \(C={\mathbf P}(R)\)). The result is applied to the generalized chain geometry given by a field in the ring \(R\).
Reviewer: F.Knüppel (Kiel)

MSC:

51B05 General theory of nonlinear incidence geometry
51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
17C50 Jordan structures associated with other structures
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