*(English)*Zbl 0880.15016

The authors consider the matrix equation ${\sum}_{i=0}^{k}{A}^{i}X{B}_{i}=C$, where $A$ is square, $X,{B}_{0},\cdots ,C$ are of such sizes that the equation makes sense, and the coefficients of all matrices are in a ring $R$ with 1. Let $B\left(\lambda \right)={\sum}_{i=0}^{k}{B}_{i}{\lambda}^{i}$, 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

are equivalent over $R\left[\lambda \right]$. 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 *W. E. Roth* [Proc. Am. Math. Soc. 3, 392-396 (1952; Zbl 0047.01901)] deals with the equation $AX-XB=C$ over a field.