When you’re tugging on the ring, there are two forces in play: you trying to make the ring slide, and friction trying to keep it in place. If friction > the force you’re applying, the ring doesn’t move.
Soap is a lubricant, which reduces the friction between the ring and your finger. That makes it more likely that friction < the force you’re applying, so the ring begins to move. Once it starts moving, friction is typically lower, so if it starts to move it will typically keep moving.
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