Would connecting propeller blades with a continuous ring reduce induced drag?
What you're talking about does exist, they are called Q-Tip Propellers.

Remember that a propeller blade is just an airfoil - like a wing - and the basic aerodynamics are no different than a wing. But the rotation of the blade creates more phenomena than a wing, in particular the helicoidal vortex one sees behind the prop and causes all sorts of propeller effects.  

In theory, nothing would prevent us from having winglets at the prop tips :
the advantages would be


Making the prop more efficient by reducing the induced drag (same as a winglet on a wing )
Reducing noise
Keeping the propeller tip speed subsonic by decreasing its length


The big problem is in aerodynamic stresses, and as far as I know there have been some quite spectacular failures during testing, so the solution is now to give a greater sweep to the tips (see that as equivalent to the 777 wing tip compared to the 787 for instance). Try and find articles on the Hartzell Q-tip.

As naval propellers are wider and capable of deealing with bigger torque stresses, modern ones do have winglets. You could find some pictures on the web.