If there is a ring homomorphism that is also an isomorphism for the underlying additive group, is it a ring isomorphism? And is this essentially just saying that when the map is closed under multiplication, the ring inherits its isomorphism from the underlying additive group?
It must be true, since the map is bijective over the sets; but it seems a bit too simple and I can't help but think I am missing a detail.
As with groups, it suffices to say that a ring homomorphism is injective. Then in either category, Grp or Ring, you get an isomorphism onto the image.
As an isomorphism on the underlying group of a ring is necessarily bijective, the answer is yes.