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The extension of Roth’s theorem for matrix equations over a ring. (English) Zbl 0880.15016
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 $$\pmatrix \lambda I-A &-C\\ 0&B(\lambda)\endpmatrix \qquad\text{and}\qquad \pmatrix \lambda I-A &0\\ 0&B(\lambda) \endpmatrix$$ 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 {\it W. E. Roth} [Proc. Am. Math. Soc. 3, 392-396 (1952; Zbl 0047.01901)] deals with the equation $AX- XB=C$ over a field.

##### MSC:
 15A24 Matrix equations and identities 15A54 Matrices over function rings
##### Keywords:
equivalence; Roth’s similarity theorem; matrix equation
Full Text:
##### References:
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