It's a slight force fit (if that's the right term) into place. So after the small screws out, on the inside, lay a short piece of hardwood dowel up against the ring. Make the dowel as large in dia as necessary to catch the ring. But short enough that you can place another piece of wood behind it and tap on it.
An EMPTY case placed into the chamber ring can help support the dowel if you don't have one laying around that's either large enough in dia or just need some more area to help dislodge it when you tap against it.
Just don't damage the ring and it's edged with a metal punch. It's actually the first 1/4" of the shell chamber itself.

The shorter the dowel,,the better the angle will be for the second piece to strike it.
It's not a perfect 'tool', but the ring isn't keyed or threaded in place. Just stuck in there pretty good usually.

Then tap the dowel forward and drive the ring out.

The two piece approach is kind of awkward but necessary as you can't get at the ring directly from the back.

I guess you could make up some kind of puller gizmo to use from the from of the frame,,but a stub of a dowel and another scrap of wood seem to work well.

When replacing,,make sure you have the thing lined up perfectly before tapping it back down into place. It's (near) impossible to adjust it to line the screw hole up once in place.
If you need to, you just use the same method to loosen it again and you get better at it for the practice.

Works the same on a 97.