Definition 10.5.1. Let $R$ be a ring. Let $M$ be an $R$-module.

We say $M$ is a

*finite $R$-module*, or a*finitely generated $R$-module*if there exist $n \in \mathbf{N}$ and $x_1, \ldots , x_ n \in M$ such that every element of $M$ is an $R$-linear combination of the $x_ i$. Equivalently, this means there exists a surjection $R^{\oplus n} \to M$ for some $n \in \mathbf{N}$.We say $M$ is a

*finitely presented $R$-module*or an*$R$-module of finite presentation*if there exist integers $n, m \in \mathbf{N}$ and an exact sequence\[ R^{\oplus m} \longrightarrow R^{\oplus n} \longrightarrow M \longrightarrow 0 \]

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