In my book (Jantzen, Algebra, 2014), Noetherian rings are defined by three equivalent conditions. I wonder how the first two can be equivalent:
Every ascending chain of ideals $(a_1) \subset (a_2) \subset \ldots$ becomes stationary. Every set of of ideals, ordered by set inclusion, has a maximum element.
The proof is just a sentence, my problem is: Isn't the set in the ascending chain condition countable while the other set is not? The proof says, suppose there is a non empty set without maximum element, then we can construct a chain of ideals that is ascending, but does not become stationary (- I see why).
Is the Axiom of Choice used to construct the chain? How could you build such a chain when your set in the second condition is uncountable?