The situation of a bubble in water is comparable to a balloon. The balloon surface is elastic. The tension of it tries to minimize the surface: if you don’t tie a knot in the balloon after blowing it up, air escapes and the surface of the balloon is minimized to the initial unstretched situation.
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Bubbles do not turn into rings naturally. Something has to be done for that. However, they have long lives and often make it up to the surface. Hence they are stable structures.
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Dolphins create bubble rings by blowing air in a water vortex ring: by flipping a fin they create a vortex ring of water. The then blow air in the ring, which goes to the center of the vortex ring. In the water vortex ring the natural location of the air is in the center of the vortex. When air and water move in a circular path like they do in the vortex ring, air and water are separated due to the centripetal force. Since density of water is larger than air, water moves at the outside, while the air ends up in the middle.
Follow the link for much more on the physics of bubble rings.
Related: Colored Bubbles – Dolphins Using Tools to Hunt – Do Dolphins Sleep? – posts on animals


