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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.

15A24Matrix equations and identities
15A54Matrices over function rings
Full Text: DOI
[1] Roth, W.: The equations AX - YB = c and AX - XB = C in matrices. Proc. amer. Math. soc. 3, 392-396 (1952) · Zbl 0047.01901
[2] Guralnick, R.: Roth’s theorems and decomposition of modules. Linear algebra appl. 39, 155-165 (1981) · Zbl 0468.16022
[3] Guralnick, R.: Matrix equivalence and isomorphism of modules. Linear algebra appl. 43, 125-136 (1982) · Zbl 0493.16015
[4] Gustafson, W.; Zelmanowitz, J.: On matrix equivalence and matrix equation. Linear algebra appl. 27, 219-224 (1979) · Zbl 0419.15009
[5] Huang, Liping; Zeng, Qingguang: The matrix equation AXB + CYD = E over a simple Artinian ring. Linear and multilinear algebra 38, 225-232 (1995) · Zbl 0824.15015
[6] Huang, Liping: The matrix equation AXB - GXD = E over the quaternion field. Linear algebra appl. 234, 197-208 (1996) · Zbl 0840.15017
[7] Cohn, P. M.: Skew field constructions. (1977) · Zbl 0355.16009
[8] Cohn, P. M.: 2nd ed. Algebra. Algebra 1 (1978)
[9] Wimmer, H. K.: The matrix equation X - AXB = C and an analogue of Roth’s theorem. Linear algebra appl. 109, 145-147 (1988) · Zbl 0656.15005