Über die multiplikative Translationsgleichung und idempotente Potenzreihenvektoren
The author proves the following theorem: Let R be a ring such that for each positive integer n the following property holds: for every idempotent matrix \(A\in M_ n(R)\) there exists an invertible matrix \(C\in M_ n(R)\) such that \(C^{-1}AC=D\) is diagonal of the form \(D=diag\{e_ 1,...,e_ n\}\) with \(e_ 1,...,e_ n\in \{0,1\}\); then every solution \(\Phi (x,z)\in Rx,z^ n\) \((Rx,z^ n=Rx,z_ 1,...,z_ n^ n\) is the n-fold cartesian product of the rings of formal power series in the variables \(x,z_ 1,...,z_ n\) with coefficients belonging to R) of the equation \(\Phi (x,\Phi (y,z))=\Phi (x\cdot y,z)\) with \(\Phi (0,0)=0\in R^ n\) has the form \(\Phi (x,z)=T^{-1}(A(x)\cdot T(z))\) where \(T(z)\in Rz^ n\) is invertible and \(A(x)\in M_ n(Rx)\) satisfies the functional equation \(A(x)A(y)=A(x\cdot y)\). This generalizes previous results of the author [ibid. 25, 233-246 (1982; Zbl 0526.39009)] and of \textit{L. Reich} [Ber. Math.-Stat. Sekt. Forschungszent. Graz 159, 22 p. (1981; Zbl 0474.30021)].
functional equations for formal power series
scientific article; zbMATH DE number 3910960