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.)
Best Answer
With your definition of Noetherian, then you don't need full AC to carry out the argument, but only the weaker principle known as Dependent Choices (DC), which asserts that one can make countably many choices in succession. In your argument, if there is no maximal ideal, then by DC you could successively pick larger and larger ones, violating the Noetherian property.