I am trying to understand the proof of Theorem 1 from [this](http://weisfeiler.com/boris/papers/1976-indag.pdf) paper V. Kac and B. Weisfeiler (Indag. Math. 1976, [DOI link](https://doi.org/10.1016/0003-4916(75)90068-8)).



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> ***Theorem 1.** Let either $p\neq 2$ or $\varrho\in X(\mathscr{T})$. Then $\gamma(Z^\mathscr{G})=U(T)^W$ and $\gamma:Z^\mathscr{G}\to U(T)^W$ is a ring isomorphism.* 
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Here $Z$ is the center of universal enveloping algebra $U(G)$ (here $G = \operatorname{Lie} \mathscr{G}$ ). $T$ is a Lie algebra of a maximal torus. $Z^{ \mathscr{G} }$ denote invariants (with respect to Ad-action of group $\mathscr{G}$) in $Z$.


Proof is not very short. But the last two lines are these



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> *From 4.4, 4.6 and 5.3 we get $\overline{U(T)^W}=\overline{\gamma(Z^\mathscr{G})}$. Since both $U(T)^W$ and $\gamma(Z^\mathscr{G})$ are integrally closed we get $U(T)^W=\gamma(Z^\mathscr{G})$ as desired.*
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Here $\bar{A}$ means field of fraction of ring $A$. Suppose I believe that these fields are isomorphic and these rings are integrally closed. 



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> **Question** How does it imply isomorphism of initial rings?
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For example $ \mathbb{k} [x, xy] \subset \mathbb{k} [x, y]$



Harish-Chandra isomorphism for characteristic $p$
In the previous lines it is proven that $U(T)^W$ is integral over $\gamma(Z^{\mathcal G})$. Since $U(T)^W\subset Frac(U(T)^W)=Frac(\gamma(Z^{\mathcal G}))$ the equality follows from $\gamma(Z^{\mathcal G})$ being integrally closed.


