The extension of Roth's theorem for matrix equations over a ring
The authors consider the matrix equation \(\sum_{i=0}^k A^iXB_i=C\), where \(A\) is square, \(X, B_0,\dots, C\) are of such sizes that the equation makes sense, and the coefficients of all matrices are in a ring \(R\) with 1. Let \(B(\lambda)= \sum_{i=0}^k B_i\lambda^i\), where \(\lambda\) is an indeterminate over \(R\). The main result (Theorem 2) is that, under suitable conditions, the above equation has a solution \(X\) if, and only if, the matrices \[ \begin{pmatrix} \lambda I-A &-C\\ 0&B(\lambda)\end{pmatrix} \qquad\text{and}\qquad \begin{pmatrix} \lambda I-A &0\\ 0&B(\lambda) \end{pmatrix} \] are equivalent over \(R[\lambda]\). The conditions under which the theorem is established are that either \(R\) is a finitely generated module over its centre \(Z\) or that \(R\) is a division ring and \(A\) satisfies a polynomial equation over \(Z\). The original theorem of \textit{W. E. Roth} [Proc. Am. Math. Soc. 3, 392-396 (1952; Zbl 0047.01901)] deals with the equation \(AX- XB=C\) over a field.
scientific article; zbMATH DE number 1044236