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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 i=0 k A i XB i =C, where A is square, X,B 0 ,,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(λ)= i=0 k B i λ i , where λ 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[λ]. 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.

15A24Matrix equations and identities
15A54Matrices over function rings