"maximal condition" means if any non-empty collection of ideals in R has a maximal element (under set inclusion). And we define noetherian ring to be a ring such that any ascending chain of ideals is finite.
I don't see why the definition implies the maximal condition. Let's suppose $S=\{\mathfrak{A}_1, \mathfrak{A}_2\}$ such that $\mathfrak{A}_1\bigcap\mathfrak{A}_2=\{0_R\}$ and none of them is unit ideal. Then there is no such maximal element. There is no need for an ideal properly contains in another ideal if it is not a maximal element. Can anyone point out what's the problem here? Thanks in advance
Edit: What I am confusing is what will be the maximal element in my example.
My definition for maximal element is: given $(S, R)$ a poset and $T\subset S$, then $s\in S$ a maximal element of $T$ if $sRy$ with $y\in S$ implies $s=y$.
And I proved "noetherian ring satisfies the maximal condition" in this way: Since any non-empty collection of ideals is a poset, let $S$ be one and $T$ be a chain of $S$. As R is noetherian, there must be an "end" for the chain, namely $\mathfrak{A}\in T$. Then it is an upper bound of $T$ and hence $S$ is inductive and the result follow from Zorn's Lemma. But then I found I can't tell what is the maximal element in my example. So either my proof is false or something else goes wrong
Suppose $S$ is a nonempty set of ideals with no maximal element.
If $I_0\in S$, there is $I_1\in S$ with $I_1\supsetneq I_0$. But neither $I_1$ is maximal, so…
This starts a recursion, and proves the ring is not noetherian.
More formally, suppose we have found a chain of ideals belonging to $S$, $I_0\subsetneq I_1\subsetneq \dots \subsetneq I_k$. Then, as $I_k$ is not maximal by assumption, we can further extend the chain. Thus $R$ is not noetherian.
Your argument is good as well: chains in $S$ are finite, so every chain has an upper bound, namely the terminal element. Zorn's lemma provides the required maximal element. But you have to remember that Zorn's lemma is a “purely existential” statement and offers no method/algorithm to determine a maximal element.