On the stabilizer of companion matrices (Q1320654)

Throughout, \(R\) denotes a commutative ring with identity, \(M_{n \times n} (R)\) the ring of \(n \times n\) matrices with entries in \(R\), and \(R[x]\) the ring of polynomials in one indeterminant \(x\) over \(R\). Let \(f(x) = x^ n - \sum^{n - 1}_{i = 1} b_ i x^ i \in R[x]\), and let \(C(f)\) denote the companion matrix of \(f(x)\) defined by \[ C(f) = \left( \begin{matrix} 0 & 0 & \cdots & 0 & b_ 0 \\ 1 & 0& \cdots & 0 & b_ 1 \\ \vdots & \vdots & & \vdots & \vdots \\ 0 & 0& \cdots & 1 & b_{n - 1} \end{matrix} \right). \] The aim of this paper is to prove the following theorem: If \(n \geq 2\), then, for \(A = (a_{ij}) \in M_{n \times n} (R)\), \(A\) commutes with \(C(f)\) if and only if \(a_{1j} = b_ 0 a_{n,j - 1}\) and \(a_{ij} = a_{i - 1,j - 1} + b_{i - 1}a_{n,j - 1}\) for all \(2 \leq i\), \(j \leq n\).