Extending morphisms with Zorn's lemma

Question

I have stumbled upon a remarkable similarity between the proof of Baer's criterion and an extension theorem in field theory. Here are the statements:

Baer's criterion: Let $R$ be a ring. A left $R$-module $E$ is injective iff every $R$-map $f:I \to E$, where $I$ is a left ideal of $R$, can be extended to a map $R\to E$.

Extension theorem: let $F\subset K$ be an algebraic field extension, and $L$ an algebraically closed field. Then every field homomorphism $\sigma: F \to L$ can be extended to a field homomorphism $K\to L$.

Now, these two theorems don't seem to have anything to do with each other, but the proofs are strikingly similar. Here's how they go:

For Baer's criterion (the non-trivial statement, i.e. the sufficiency of the ideal condition): to prove injectivity, we let $A\subset B$ be a submodule and $f:A\to E$ an $R$-map, and define $X=\{(A',g'): A\subset A'\subset B,\, g': A' \to E,\, g'|_A=f\}$.

Let $\leq$ be the partial order in $X$ given by $(A',g')\leq (A'',g'') \iff A'\subset A'' \text{ and } g''|_{A'}=g'$.

By Zorn's lemma there's $(A_0,g_0)$ a maximal element. If $A_0=B$ we're done, if not, let $b\in B\setminus A_0$, define $I=\{r\in R: rb\in A_0\}$ and $h:I\to E$, $r \mapsto g_0(rb)$. Apply the hypothesis to find an extension $\tilde{h}$ of $h$ to $R$. Let $A_1=A_0 + Rb$ and $g_1:A_1 \to E$, $a_0+rb \mapsto g_0(a_0)+r\tilde{h}(1)$, and $(A_1,g_1)$ contradict the maximality of $(A_0,g_0)$.

For the extension theorem: define $M=\{(A,\tau):F\subset A \subset K,\, \tau:A\to L,\, \tau|_F=\sigma\}$. Let $\leq$ be the partial order on $M$ given by $(A,\tau)\leq (A',\tau') \iff A\subset A' \text{ and } \tau'|_A=\tau$.

By Zorn's lemma there's $(A,\tau)$ a maximal element. If $A=K$ we're done, if not let $\alpha \in K\setminus A$, I claim $\tau$ extends to $A(\alpha)\to L$, contradiction.

Let $p$ be the minimal polynomial for $\alpha$ over $A$. The polynomial $\tau p\in L[X]$ has a root $r\in L$ by hypothesis. Since $A(\alpha)= \frac{A[X]}{\langle p \rangle}$, define

$\tilde{\tau}: \frac{A[X]}{\langle p \rangle} \to L$ as $\tilde{\tau}|_A=\tau$, $\tilde{\tau}(X)=r$, and $(A(\alpha),\tilde{\tau})$ contradicts the maximality of $(A,\tau)$.

The resemblance of both proofs shouldn't come as such a surprise since both are about extending morphisms, so maybe the only observation to make is "it's a useful technique, remember it's useful for extending morphisms". If that's it, then this question is useless. But I'm intrigued. Is this technique used to prove other extending theorems? Is it possible to generalize it and write a single categorical proof for situations of this kind? Any other observations are appreciated (or perhaps it's just a dumb observation that doesn't serve any purpose).

Answer 1

I suspect you'll find highly interesting the following paper on related topics, which I mentioned in some prior answers that you may find of interest. Below is the introduction of said paper.

Answer 2

I do see some nontrivial commonality here, but it is not precisely between the two theorems you suggest.

Namely, just as the injective modules are (obviously!) the injective objects in the category of $R$-modules, the algebraically closed fields are the injective objects in the category of fields. Indeed, the latter statement is precisely your Extension Theorem.

Thus your Extension Theorem doesn't strike me as a "Baer criterion" per se. Perhaps it is better the other way around: if we define an injective field as being a field which satisfies the conclusion of the Extension Theorem, then maybe "Baer's Criterion" is that a field is injective iff it is algebraically closed? (To be clear, this is definitely a true fact; it's not clear whether it is worthy of this name.)

Let me try this: to check whether a field $K$ is injective, it suffices to check that for every maximal ideal $I$ of $K[t]$ there exists a $K$-algebra map $K[t]/I \rightarrow K$. This is perhaps "Baeresque"?

Note that I came to the analogy here by trying to understand why the injective envelope $M \rightarrow E(M)$ is not a natural [in the categorical sense] construction: it is unique up to nonunique isomorphism over $M$. This is just the same situation as the algebraic closure of a field. And then I found this paper by Adamek, Herrlich, Rosicky and Tholen exploring this issue in a more general categorical context.