I am taking a ring theory class (the class is online…), and we just proved that every ring has a maximal ideal. Anyone who is familiar with the proof knows that in order to prove it, we have to use Zorn's lemma. Basically proofs that are using Zorn's lemma have the same "plan" of the proof, for example, that is the proof:
Suppose $R$ is a ring (with unit) and $I \triangleleft R$ is a proper ideal.
Let $S$ be the set that contains $I$ and all the ideals $K$ such that $I\subseteq K$.
Let $C$ be a chain, and now we can define $J=\bigcup_{I\in C}I$. It is easy to verify that $J$ is an ideal, and because each ideal in the union doesn't include $1_R$ , $J$ doesn't include $1_R$ as well, thus, $J$ is also a proper ideal ($J\in S$). So we have found an upper bound for each chain, and now, by Zorn's lemma we can conclude that $S$ has a maximal element, (which is a maximal ideal).
So, my question is: why do we need Zorn's lemma? why isn't the union a maximal element? also, in general, if we find an upper bound like that in other proofs similar to this one ('every vector space has a basis' for example) , why it is not a maximal element?
Best Answer
The union is simply an upper bound of the chain. What if the chain had one element, say $\{I\}$? The union of this chain is just $I$. So why is $I$ a maximal element now?
The only condition on the chain itself which ensures that its upper bound is a maximal element is to require that it is maximal. But now you're working around yourself: requiring that every partial order has a maximal chain is equivalent to Zorn's lemma.
What Zorn's lemma ensures is that by checking a local condition (i.e. that chains have upper bounds), we can derive a global property of the partial order (i.e. the existence of a maximal element).
If you want to understand better the use of Zorn's lemma, I suggest thinking about the Teichmüller–Tukey lemma instead:
We say that a family of sets $\scr F$ has finite character if $A\in\scr F$ if and only if every finite $A_0\subseteq A$ is also in $\scr F$.
To see why this is "the usual use of Zorn's lemma", note that the standard appeal to Zorn's lemma is in the case where a certain property is of finite character. For example, in the case vector space bases, being linearly independent is a property that has finite character: if a set is not linearly independent, there is a finite subset witnessing that.
The only reason to use Zorn's lemma is that sometimes it's slightly simpler. Like in the case that you want to extend an ideal to a maximal ideal, or even extend a given linearly independent set to a basis. But the idea stays the same, we use the finite character of our property.