## 15.53 Abelian categories of modules

Let $R$ be a ring. The category $\text{Mod}_ R$ of $R$-modules is an abelian category. Here are some examples of subcategories of $\text{Mod}_ R$ which are abelian (we use the terminology introduced in Homology, Definition 12.10.1 as well as Homology, Lemmas 12.10.2 and 12.10.3):

1. The category of coherent $R$-modules is a weak Serre subcategory of $\text{Mod}_ R$. This follows from Algebra, Lemma 10.90.3.

2. Let $S \subset R$ be a multiplicative subset. The full subcategory consisting of $R$-modules $M$ such that multiplication by $s \in S$ is an isomorphism on $M$ is a Serre subcategory of $\text{Mod}_ R$. This follows from Algebra, Lemma 10.9.5.

3. Let $I \subset R$ be a finitely generated ideal. The full subcategory of $I$-power torsion modules is a Serre subcategory of $\text{Mod}_ R$. See Lemma 15.88.5.

4. In some texts a torsion module is defined as a module $M$ such that for all $x \in M$ there exists a nonzerodivisor $f \in R$ such that $fx = 0$. The full subcategory of torsion modules is a Serre subcategory of $\text{Mod}_ R$.

5. If $R$ is not Noetherian, then the category $\text{Mod}^{fg}_ R$ of finitely generated $R$-modules is not abelian. Namely, if $I \subset R$ is a non-finitely generated ideal, then the map $R \to R/I$ does not have a kernel in $\text{Mod}^{fg}_ R$.

6. If $R$ is Noetherian, then coherent $R$-modules agree with finitely generated (i.e., finite) $R$-modules, see Algebra, Lemmas 10.90.5, 10.90.4, and 10.31.4. Hence $\text{Mod}^{fg}_ R$ is abelian by (1) above, but in fact,in this case the category $\text{Mod}_ R^{fg}$ is a (strong) Serre subcategory of $\text{Mod}_ R$.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AZ5. Beware of the difference between the letter 'O' and the digit '0'.