Doubly coprime factorizations (DCF) play an important role in the design of compensators for lumped linear time invariant systems. They are defined as stable proper rational matrices, and they not only constitute a starting point for the fractional approach to all stabilizing compensators, but they are also closely related to the frequency domain design of (optimal) state feedback and estimation, and of the resulting observer-based compensators. When defining DCFs related to reduced-order observers, some rational matrices become improper. This can be prevented by the introduction of artificial (stable) dynamics-the identity elements-which cancel in the system and in the compensator transfer matrices. Such pole-zero cancellations, however, are contrary to the usual notion of coprimeness. It is shown here that these additional dynamics, which have no meaning in an observer-based compensator scheme are superfluous, and that the parametrization of all stabilizing compensators is feasible on the basis of possibly nonproper factorizations not containing identity elements. For such factorizations, the new definition of strictly doubly coprime factorizations (SDCF) is proposed.