Another interesting fact about fields is that a commutative ring is a field if and only if every ideal is a prime ideal. (Obviously, every ideal of a field is a prime ideal. The converse is more interesting...)
This is a great example of one of those things in abstract maths that is hard to follow when you learn it, but once you’ve been through it a few times and learnt the definitions to heart it’s really just a rephrasing of the definitions.
FYI many sources do not count the entire ring as an ideal. If you do, you’d have to define “maximal ideal” to mean “an ideal that is maximal with respect to inclusion, ignoring the entire ring ideal.”