The authors consider the matrix equation , where is square, are of such sizes that the equation makes sense, and the coefficients of all matrices are in a ring with 1. Let , where is an indeterminate over . The main result (Theorem 2) is that, under suitable conditions, the above equation has a solution if, and only if, the matrices
are equivalent over . The conditions under which the theorem is established are that either is a finitely generated module over its centre or that is a division ring and satisfies a polynomial equation over . The original theorem of W. E. Roth [Proc. Am. Math. Soc. 3, 392-396 (1952; Zbl 0047.01901)] deals with the equation over a field.