[[R-monoid]]
# Module-finite $R$-monoid

An $R$-monoid (or [[ring extension]]) $T$ is called **module-finite**[^term] iff $T$ is [[Finitely generated module|finitely generated]] as an $R$-module, #m/def/ring 
i.e. there exists an onto $R$-[[module homomorphism]]
$$
\begin{align*}
R^{(n)} \twoheadrightarrow T
\end{align*}
$$
for some $n \in \mathbb{N}_{0}$.
This should not be confused with the weaker condition of [[R-monoid of finite type]].[^2009]

  [^2009]: 2009\. [[Sources/@aluffiAlgebraChapter02009|Algebra: Chapter 0]], §III.6.5, p. 171
  [^term]: The usual terminology is just **finite**, but I find this misleading.

## Properties

1. [[Finitely generated module over a module-finite R-ring]]

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