It is known that any ring A (say commutative with 1) has a maximal ideal. The proof uses Zorn's lemma.
Now I heard some people saying that if we assume A to be noetherian, then we don't need to use Zorn's lemma. The argument would basically be as follows:
"Suppose it doesn't have a maximal ideal. Then we can build an ascending chain of distinct ideals."
But, as far as I know it, we have to use Zorn's lemma in order to construct such an ascending chain. Am I right?
If I am right, is it still true (via some other argument) that we don't need to use Zorn's lemma to prove the result?
(EDIT: My definition of noetherian ring is that any ascending chain of ideals stabilizes.)