how to half shapes? on illustrator

Sam,

Or in any (older) version, regardless of whether you have a full circle or a circular ring (Compound Path), to get a (set of) simple path(s), you may ClickDrag between opposite Anchor Points with the Line Segment Tool and Object>Path>Divide Objects Below, then Ctrl/Cmd+Shift+G to Ungroup, then delete/hide (invisibilize)/move/whatever the unwanted (other half) part of the shape.

Please include you Adobe Illustrator version when posting, and screesnhots if that can help explain what you are trying to do.

Since CC2015 you have shapes and can do this easily and precisely by changing one number, if this was built by drawing 2 circles and make compound path or similar.

You can also create  Mask, consult the Adobe help on them.

If you want to cut a shape in half and retain both halves, you can use the Knife Tool.