Zorn’s Lemma: If is a partially ordered set with the property that every totally ordered subset (often called a chain) has an upper bound, then there exists a maximal . (that is, for , we have )
This “lemma” is a basic tool for dealing with large collections in a systematic fashion. (if you are interested in the background behind it, see the Wikipedia articles on Zorn’s Lemma and the axiom of choice) Here I’ll demonstrate a fun application in module theory, but there are hundreds of other examples throughout algebra and set theory.
If is a commutative ring, it turns out that there exist -modules with a fascinating property: If is an -module homomorphism and , where and are -modules, then there exists an -module homomorphism extending . Such modules are called injective modules.
This is a surprising property, because in general we cannot make functions “bigger”. For example, if , we have a map taking , but in any map from we have .
It is not at all obvious at first that injective modules exist, (other than the zero module) let alone what they look like. Nevertheless, there is one very simple example: the -module . Intuitively, we can imagine dividing by elements of to extend the domain of some function, but even in this case there is some difficulty in proving that is injective. We need, as they say, to “develop some theory”.
Theorem: (Baer’s Criterion) Suppose that has the property that if is any ideal of and is an -module homomorphism, there exists an -module homomorphism extending . Then is an injective -module.
This is a powerful theorem because it is much easier to describe the ideals of a ring than it is to describe every possible homomorphism from every possible -module. For and , we see quite quickly that any map can be extended to , so this theorem shows that is injective.
But how to prove Baer’s Criterion? It helps to start by considering the simplest cases.
For a basic example, suppose that is an inclusion of cyclic modules, (that is, modules generated by a single element) and is a homomorphism. We will show that we can extend to . Let . Then the map defined by is a homomorphism, so we can extend to , by the hypothesis. Define . Some basic manipulation of the definitions show that is a well-defined homomorphism that extends .
Unfortunately, modules are not, in general, cyclic. But if for some , and we have , we can perform a nearly identical construction. Let , and extend to . Then we can define . Again, it is easy to show that is a well-defined homomorphism that extends . (this is actually an elementary kind of pushout)
This shows that we can always increase the domain of by adding a new element. In particular, this proves Baer’s criterion for finitely generated modules. But how can we extend to argument to the infinitely generated case? (for example, the -module )
This is where Zorn’s Lemma comes in. Say that and is a homomorphism. We can form the partially ordered set of extensions of . (where if and extends ) Then if is a totally ordered set, we can form , so this verifies the chain condition. Hence there exists a maximal , by Zorn’s Lemma.
But if we have some , we can extend to , contradicting the maximality of . So we must have , and we are done.
Notice that Zorn’s lemma takes the place of a wishful induction. We want to say, “We keep adding elements to the domain until we are done,” but induction alone is not powerful enough to make this rigorous.