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.)

finitefamilies of non-empty sets (or finite sequences from infinite sets). But that's all you'll ever need for a Noetherian ring before you reach a maximal ideal." $\endgroup$ – David Feldman Jan 28 '11 at 0:35