Remark 9.2.0.3. Let $R$ be an associative ring and let $\operatorname{\mathcal{C}}$ be the category of left $R$-modules. Proposition 9.2.0.1 asserts that an object $M \in \operatorname{\mathcal{C}}$ is compact if and only if it is finitely presented as an $R$-module. This result has counterparts for many other classes of mathematical structures, which can be proved by the same argument:
Let $\operatorname{\mathcal{C}}= \operatorname{Group}$ be the category of groups. An object $G \in \operatorname{\mathcal{C}}$ is compact if and only if it is finitely presented as a group: that is, it can be realized as the quotient of a finitely generated free group by the normal subgroup generated by finitely many elements.
Let $\operatorname{\mathcal{C}}= \mathrm{Ring}$ be the category of associative rings. Then an object $R \in \operatorname{\mathcal{C}}$ is compact if and only if it is finitely presented as a ring: that is, it can be realized as the quotient of a noncommutative polynomial ring $\operatorname{\mathbf{Z}}\langle x_1, x_2, \cdots , x_ n \rangle $ by a finitely generated two-sided ideal.
Let $\operatorname{\mathcal{C}}= \mathrm{CRing}$ be the category of commutative rings. Then an object $R \in \operatorname{\mathcal{C}}$ is compact if and only if it is finitely presented as a commutative ring: that is, it can be realized as the quotient of a finitely generated polynomial ring $\operatorname{\mathbf{Z}}[x_1, \cdots , x_ n]$ by an ideal $I$ (here $I$ is automatically finitely generated, by virtue of the Hilbert basis theorem).