process to pass from the given category to an associated category E with a Noetherian ring $\mathbb E$. This is a general categorical argument.

To unify notation, we agree that $\mathbb S$ may also take the value $[0,1]$, and that $T_{[0,1]}$ is the monad of finitely supported subprobability distributions giving rise to the category .

  $\mathbb S$   $\mathbb N$   $\mathbb Q_+$   $\mathbb R_+$   $[0,1]$
  ------------- ------------- --------------- --------------- ---------
                N ()                          ()              
  E             Z ()          Q (Q)           R (R)           R (R)

For the formulation of the extension lemma, recall that the starting category is the Eilenberg-Moore category of the monad ${T_{\mathbb{S}}}$ and the target category E is the Eilenberg-Moore category of ${T_{\mathbb{E}}}$. We write $\eta_{\mathbb S}$ and $\mu_{\mathbb S}$ for the unit and multiplication of ${T_{\mathbb{S}}}$ and analogously for ${T_{\mathbb{E}}}$. We have ${T_{\mathbb{S}}}\le {T_{\mathbb{E}}}$, via the inclusion monad morphism $\iota\colon {T_{\mathbb{S}}}\Rightarrow {T_{\mathbb{E}}}$ given by $\iota_X(u) = u$, as $\eta_{\mathbb E} = \iota {\mathrel{\circ}}\eta_{\mathbb S}$ and $\mu_{\mathbb E} {\mathrel{\circ}}\iota\iota =\iota{\mathrel{\circ}}\mu_{\mathbb S}$ where $\iota\iota \stackrel{\text{def}}{=}{T_{\mathbb{E}}}\iota{\mathrel{\circ}}\iota \stackrel{\text{nat.}}{=} \iota {\mathrel{\circ}}{T_{\mathbb{S}}}\iota$. Recall that a monad morphism $\iota\colon T_\mathbb S \to T_\mathbb E$ defines a functor $M_\iota\colon {{\text{\tt{Set}}}^{T_\mathbb E}} \to {{\text{\tt{Set}}}^{T_\mathbb S}}$ which maps a $T_\mathbb E$-algebra $(X, \alpha_X)$ to $(X, \iota_X{\mathrel{\circ}}\alpha_X)$ and is identity on 