ROTHS THEOREMS OVER COMMUTATIVE RINGS

In 1952, W.E. Roth showed that matrix equations of the forms AX-YB = C and AX-XB = C over fields can be solved if and only if certain block matrices built from A, B, and C are equivalent or similar. We show here that these criteria remain valid over arbitrary commutative rings. To do this, we use standard commutative algebra methods to reduce to the case of Artinian rings, where a simple argument with. © 1979.