That would be because there is an electric field inside the battery and outside it--the ∫E.dl around a loop of which will be zero.
Yes they would. But I used the battery as an example. The original question did not have it. I used the example of an isolated battery to bring out a certain point.
The electric field produced by the battery and the induced field do not add to zero but they do produce a static electric field by causing charges to be accumulated at the ends of the ring.That is like a capacitor. That field stops further accumulation of charges and gives us a finite potential.
The ∫E.dl is -dφ/dt but that happens over a cycle. The loop is incomplete and hence no area or flux can be associated with the integral.
I am sorry but I have trouble putting my points across.