nLab
Baer's criterion

Idea

Baer’s criterion is a criterion for detecting injective objects in a category of modules: injective modules.

Statemen

Let $R$ be a commutative ring and $C=R$Mod the category of $R$-modules.

Proposition

(Baer's criterion)

An object $Q\in R\mathrm{Mod}$ is injective precisely if for $I$ any left $R$-ideal regarded as an $R$-module, any morphism $g:I\to Q$ in $C$ can be extended to all of $R$ along the inclusion $I\hookrightarrow R$.

Sketch of proof

Let $i:M\hookrightarrow N$ be a mono in $R\mathrm{Mod}$, and let $f:M\to Q$ be a map. We must extend $f$ to a map $h:N\to Q$. Consider the poset whose elements are pairs $(M\prime ,f\prime )$ where $M\prime$ is an intermediate submodule between $M$ and $N$ and $f\prime :M\prime \to Q$ is an extension of $f$, ordered by $(M\prime ,f\prime )\le (M″,f″)$ if $M″$ contains $M\prime$ and $f″$ extends $f\prime$. By an application of Zorn's lemma, this poset has a maximal element, say $(M\prime ,f\prime )$. Suppose $M\prime$ is not all of $N$, and let $x\in N$ be an element not in $M\prime$; we show that $f\prime$ extends to a map $M″=⟨x⟩+M\prime \to Q$, contradiction.

The set $\{r\in R:rx\in M\prime \}$ is an ideal $I$ of $R$, and we have a module map $g:I\to Q$ defined by $g(r)=f\prime (rx)$. By hypothesis, we may extend $g$ to a module map $k:R\to Q$. Writing a general element of $M″$ as $rx+y$ where $y\in M\prime$, it may be shown that

$$f″(rx+y)=k(r)+g(y)$$f''(r x + y) = k(r) + g(y)

is well-defined and extends $f\prime$, as desired.

Consequences

Corollary

Let $R$ be a Noetherian ring, and let $\{Q_j{\}}_{j\in J}$ be a collection of injective modules over $R$. Then the direct sum $Q={⨁}_{j\in J}{Q}_j$ is also injective.

Proof

By Baer’s criterion, it suffices to show that for any ideal $I$ of $R$, a module map $f:I\to Q$ extends to a map $R\to Q$. Since $R$ is Noetherian, $I$ is finitely generated as an $R$-module, say by elements $x_1,\dots ,x_n$. Let $p_j:Q\to Q_j$ be the projection, and put $f_j=p_j\circ f$. Then for each $x_i$, $f_j(x_i)$ is nonzero for only finitely many summands. Taking all of these summands together over all $i$, we see that $f$ factors through

$$\prod _{j\in J\prime }{Q}_j=\underset{j\in J\prime }{⨁}{Q}_j\hookrightarrow Q$$\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q

for some finite $J\prime \subset J$. But a product of injectives is injective, hence $f$ extends to a map $R\to {\prod }_{j\in J\prime }{Q}_j$, which completes the proof.

Conversely, a result of Bass and Papp is that $R$ is Noetherian if direct sums of injective $R$-modules are injective. See Lam, Theorem 3.46.