We introduce a concept of countably strong inclusions ${ riangleleft}$ and that of ${ riangleleft}-{sigma}$-ideals and prove that the subframe $S({ riangleleft})$ of the frame ${sigma}IdL$ of ${sigma}$-ideals is a Lindel$ddot{o}$fication of a frame L. We also deal with conditions for which the converse holds. We show that any countably approximating regular $D({aleph}_1)$ frame has the smallest countably strong inclusion and any frame which has the smallest $D({aleph}_1)$ Lindel$ddot{o}$fication is countably approximating regular $D({aleph}_1)$.