# What are examples of cogenerators in R-mod?

Fill in the blank, please :)

Let $\mathcal C$ be a complete and cocomplete abelian category. A generator in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ is a colimit of a (small) diagram made entirely of $X$s; in this way, $X$ knows everything there is to know about $\mathcal C$. When $\mathcal C$ is the category of all modules of some ring $R$, then an example of a generator is $R$, thought of as an $R$-module. A cogenerator in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ is a limit of a (small) diagram made entirely of $X$s; in this way, $X$ knows everything there is to know about $\mathcal C$. When $\mathcal C$ is the category of all modules of some ring $R$, then an example of a cogenerator is ________.

The only examples I know are: when $R = \mathbb Z$, an example of a cogenerator is the rational circle $\mathbb Q / \mathbb Z$; when $R = \mathbb K$ is a field, an example of a cogenerator is $\mathbb K$. But by some version of the Law of Small Numbers, these examples are not enough for me to see how to (or, in fact, whether it is possible to) generalize.

• How about: direct sum of injective hulls of all simple $R$-modules? In case $R$ is local Artinian, this is just the injective hull of the residue field, and is finitely generated. – Hailong Dao Sep 8 '10 at 18:54

You can say (albeit rather abstractly) what any cogenerator must look like. The following can be found in T.Y. Lam's Lectures on Modules and Rings, Theorem 19.10. Let $\{V_i\}$ be a complete set of simple right $R$-modules, with injective hulls $E(V_i)$. Then $U_0 = \bigoplus E(V_i)$ is a cogenerator, called the canonical cogenerator of $\mathrm{Mod}_R$, and any module $U_R$ is a cogenerator for $\mathrm{Mod}_R$ if and only if $U_0$ can be embedded in $U$.
Note that if $R$ is right noetherian, then the direct sum of injective right $R$-modules is again injective. Hence $U_0$ is injective, and in this case $U_R$ is a cogenerator if and only if $U_0$ embeds in $U$, if and only if $U_0$ is a direct summand of $U$.
Referring to your examples of $R$ above: If $R = \mathbb{Z}$ then $U_0 = \mathbb{Q}/\mathbb{Z}$. If $R = \mathbb{K}$ then $U_0 = \mathbb{K}$. Both of these rings are noetherian, so the previous paragraph applies. (In particular, every nonzero $\mathbb{K}$-vector space is a cogenerator).
• As Lam notes (in 19.13), if you happen to need an injective cogenerator but $U_0$ is not injective, just pick $U^0:=E(U_0)$, the minimal injective cogenerator. – Jose Brox Nov 6 '17 at 10:55
For any ring $R$, the $R$-module $Hom_{\mathbb Z}(R,\mathbb Q/\mathbb Z)$ is an injective cogenerator of the category of $R$-modules. Here $\mathbb Q/\mathbb Z$ can be replaced with $\mathbb R/\mathbb Z$ or any other injective cogenerator of the category of abelian groups. When $R$ is an algebra over a field $\mathbb K$, another example of an injective cogenerator of the category of $R$-modules is $Hom_{\mathbb K}(R,\mathbb K)$.