I've got a problem, and it's that I can't see the difference of one of the definitions of Noetherian ring, and supposing the zorn lemma for that ring.
I mean, if I use the definition of the maximal element in every non-empty set of ideals it's clear to me that if I suppose the zorn lemma as true, every ring becomes (For me) a "Weakly-noetherian ring", and I become able to prove some properties that before I only could with noetherians.
But when I use the definition of the ascending chain which terminates, I can't see the difference. I mean, if I have an ascending chain, and it terminates, it'd have a maximal element... and if it has a maximal element it should terminate. But that wouldn't be possible, because if we suppose Zorn lemma with my bold statement we would get that every ring is noetherian... and I already have seen some non-noetherian rings so...
I think there would be a mistake in the reasoning in bold... but I can't locate it...
Is there any chain which has a maximal element, and doesn't terminate? Is there something else I'm not seeing and I should?
Not every set of ideals has a maximal elements.
It is true that with the axiom of choice, the set of all ideals has a maximal element, but it is not true that every set of ideals has a maximal element.
For example, look at $\Bbb Z[\{x_n\mid n\in\Bbb N\}]$ and consider the set of ideals which are generated by $\{x_i\mid i<n\}$ for each $n\in\Bbb N$. This set of ideals has no maximal element. But the ring itself, of course, has a maximal ideal.
Since you brought up the axiom of choice and Noetherian ideals, it should be noted that the definition "every ascending chain stabilizes" is not equivalent to "every non-empty set of ideals has a maximal element" without some choice. It is in fact strictly weaker.