Yeah, best argument I can come up with is that the multiplication over the zero ring minus 0 does not form a group. But you'd need to modify the usual definition of a field for this to look natural.

There is quite a lot of research on a zero field, which shows up sometimes (never as a conceret object, but more like A_p is to the field of p elements what A_0 is to the zero field). Apparently you can't just use the zero ring, though the reasons why are beyond my pay-grade.