How do you split a ring on only one side?

I have a ring that I want to split on only one side.  However, splitting it with split lines or splitting the part using split features doesn't work.  I have tried splitting into 4 parts and then combining bodies.  I have tried splitting with a sketch, that didn't work.  Any creative ideas? I want to split the ring where the plane is with no gap.

SolidworksParts And Features