Normal physics wouldn't do the job. A planetary ring inside an atmosphere would be moving at approximately 8 km/s. This is the orbital velocity for an Earth mass planet. Atmospheric drag would soon decelerate the pieces of the planetary ring. It wouldn't last very long at all.
As a thought experiment: make a planetary ring magically appear in an atmosphere. At a guesstimate it wouldn't last one whole orbit.
Therefore, if a planetary ring does exist in atmosphere some form of magic or 'magic' must be at work. This could be a barrier to prevent atmospheric drag slowing down the ringlets and the flying islands or a constant force to maintain a constant orbital velocity.
For example, one can imagine a teleportation surface in front of the ringlets instantaneously transporting the atmosphere it impacts to somewhere behind them where it can rematerialize harmlessly. This would eliminate atmospheric drag. But some kind of force-field might necessary to maintain a breathable atmosphere for the flying islands.
In summary, if there is some form of protection to negate atmospheric drag, a form of 'magic' propulsion, if needed, to maintain velocity and a containment field to make the flying islands habitable. Then a planetary ring could orbit inside an atmosphere.
A simpler solution might to have a traditional, scientifically sound planetary ring orbiting outside the atmosphere and have the flying islands periodically rise up from the planet below to join the ring. The flying islands would spend a period of time as part of the ring and later on they would descend back down to the planet.
As has the advantage that the flying islands can be inhabited by lifeforms that evolved on the planet. There can be a form of antigravity that allows them to either move through the atmosphere or rise to join the planetary ring for a time. Again there needs to be mechanism to contain an atmosphere around the flying islands. However, this could be simply part of the antigravity that levitates the islands in the first place.
This idea is that the islands are habitable and can remain habitable while in the planetary ring. Also, they return to the planet to restock air, water, and whatever else they need to survive. They won't be permanently part of the planetary ring. This is only a suggestion.
This alternative solution does turn the original question on its head. But it does that the simplest solution would be to consider to make the flying islands that are part of a planetary ring habitable.