[[Finitely generated module]]
# Finitely generated module over a module-finite $R$-monoid

Let $T$ be a [[Module-finite R-monoid]] and $M$ be a [[finitely generated module|finitely generated]] $T$-module.
Then $M$ is a finitely generated $R$-module. #m/thm/module 

> [!check]- Prove
> Let $\{ t_{i} \}_{i=1}^m$ be an $R$-spanning set for $T$ and $\{ m_{i} \}_{i=1}^n$ be a $T$-spanning set for $M$.
> Then
> $$
> \begin{align*}
> \{ t_{i}m_{j} : i \in \mathbb{N}_{m}, j \in \mathbb{N}_{n} \}
> \end{align*}
> $$
> is an $R$-spanning set for $M$, since any $v \in M$ may be expressed as an $T$-linear combination of $m_{j}$'s and the coëfficients may then be expressed as linear combinations of $t_{i}$'s. <span class="QED"/>

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