As others have pointed out, fitting the ring to the prop will greatly increase the stress on the blades. The same effect can be had with a nicely fitted shroud.
There has indeed been a plane which used this concept, the RFB FanTrainer (see picture below). To reduce weight and wetted area, the prop diameter was much smaller than with a regular propeller, so the overall efficiency was not better. However, the smaller rotating inertias did produce a more turbine-like effect (less precession), so the concept was used for a basic trainer for future jet pilots.
In the end, the FanTrainer enjoyed only limited success and was discontinued after 50 had been built. The design was too lightweight to support all the desires of air forces for a basic trainer, and the private market at that time was shrinking and full of older planes which served the cost-conscious customers equally well. It did, however, offer almost jet-like characteristics for a uniquely low price per flight hour.
In general, if you want to shroud the propeller for better efficiency, you need to accept the higher surface area of the shroud, which will quickly add more drag than you are ever likely to save by preventing flow around the prop tips.
What could be saved by shrouding the prop? Induced drag would be the same, since this comes from lift creation. The classical theory for minimum induced loss propellers by A. Betz and L. Prandtl requires an elliptic lift distribution over the propeller disc, such that lift smoothy tapers off at the tips. Artificially increasing it would only help if this could reduce blade chord at the tips - since the tips see the highest dynamic pressure, this could indeed translate into less friction drag. However, this gain is small when compared to the massive increase in friction drag of a shroud.
At high speeds the induced losses are small, and other factors become dominant. Note that turbofans and highly loaded propellers are not designed for minimum induced loss, but for maximum thrust with a given diameter. A shrouded propeller can enjoy a higher disc loading, so you get the same thrust with smaller blades and lower tip speeds, which will help in high speed efficiency. Smaller blades translate into less friction losses on the prop, and lower tip speeds translate into higher cruise speed before Mach losses begin to bite.
Thus, at high speed a shroud can be helpful when it is not too large. Turbofan engines suffer from this dilemma. They could have much higher bypass ratios than today, but this would mean huge nacelles, and the increased nacelle drag would offset the gains from the increased bypass ratio. Actively laminarising nacelle flow is the way forward here, but so far the practical implementation has yet to happen.