The Equation $AXB + CYD = E$ over a Principal Ideal Domain

This paper considers the equation $AXB + CYD = E$ in matrices over a principal ideal domain. Under the assumption that $[ A:C ]$ is left invertible and [ B':D' ]^ is right invertible in the domain, it is shown that this equation is solvable if and only if both $AX + YD = E$ and $XB + CY = E$ are solvable. The set of all solutions to the equation are shown to be in bijective correspondence with the set of solutions to the latter two equations modulo an equivalence relation.