[[R-monoid of finite type]]
# Commutative $R$-monoid of finite type

A [[Commutative R-monoid]] $T$ being [[R-monoid of finite type|of finite type]] is equivalent to the existence of an $R$-monoid homomorphism #m/thm/calg
$$
\begin{align*}
R[x_{1},\dots,x_{n}] \twoheadrightarrow T
\end{align*}
$$
from the [[polynomial ring]] in $n$ indeterminates.

> [!check]- Proof
> Noting that the polynomial ring is the “abelianization” of the [[Free R-ring]], 
> this follows from the characterization of a general [[R-monoid of finite type]]. <span class="QED"/>

As such, these are just quotients of the [[polynomial ring]].

## Properties

- [[Hilbert's basis theorem]]

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