You could use a face to cut it as you have done (along with Intersect Faces), but because geometry sticks together in SU it’s generally easier to make a copy of something and delete half of each copy than to cut it in two. Or, start with the two halves (make a copy and delete half) as faces and extrude them after. Or start with one half, extrude it, and copy it. Really, the last thing you want to do is cut a full ring in half.
-Gully