Example 10.159.9. Lemma 10.159.8 does not work if the ring is not Noetherian. For example consider the action of $G = \{ +1, -1\}$ on $A = \mathbf{C}[x_1, x_2, x_3, \ldots ]$ where $-1$ acts by mapping $x_ i$ to $-x_ i$. The invariant ring $R = A^ G$ is the $\mathbf{C}$-algebra generated by all $x_ ix_ j$. Hence $R \subset A$ is not finite. But $R$ is a normal domain with fraction field $K = L^ G$ the $G$-invariants in the fraction field $L$ of $A$. And clearly $A$ is the integral closure of $R$ in $L$.

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