THE EXTENSION OF ROTH THEOREM FOR MATRIX EQUATIONS OVER A RING

This paper extends Roth's similarity theorem as follows: Let R be a ri ng with identity, B(lambda) = Sigma(i=0)(k)B(i) lambda(i) is an elemen t of R-mxq[lambda]. If either R is a division ring and A is an element of R-nxn is algebraic, or R is finitely generated as a module over it s center, then the matrix equation Sigma(i=0)(k)A(i)XB(i) = C over R h as a solution if and only if ((lambda I-A)(-C)(O)(B lambda)) congruent to ((lambda I-A)(O)(O)(B lambda)). (C) Elsevier Science Inc., 1997.