According to my physics book, inside a resistanceless battery (it is a part of a closed circuit) the conservative field has the same magnitude but opposite direction to the non conservative field. Thus $E^* = -E$.
$E^*$ is the non-conservative field.
Anyway, I can apply the Ohm's law to a real battery because a current flows through it.
$$J = σ(E^* + E)$$
What about the current inside an ideal battery? If we try to apply the Ohm's law (we can't do this, actually) to an ideal battery, then we find an indeterminate form, because the conductibility is infinite and the net electric field is zero. I would expect that inside an ideal battery there isn't a current flow, is it?