I have the following version of Noether Normalisation Lemma: Let $R=F[y_1,\dots,y_n]$ be a finitely generated $F$-algebra. Then there exists a subset $\{x_1,\dots,x_k\}$ of $R$ which is algebraically independent over $F$ and such that $R$ is a finitely generated $F[x_1,\dots,x_k]$-module.
The proof goes by induction on $n$ and for the induction step it suffices to show that there's a subring $A$ of $R$ generated by $n-1$ elements as a $F$-algebra such that $R$ is finitely generated as an $A$-module. Then by induction we can find $\{x_1,\dots,x_k\}\subset A$ algebraically independent over $F$ such that $A$ is a finitely generated $F[x_1,\dots,x_k]$-module. It then follows that $R$ is a finitely generated $F[x_1,\dots,x_k]$-module.
At the end of the construction of such a subring $A$, we end up with $A$ being finitely generated as a $F$-algebra and such that for each $i\in\{1,\dots,n\}$, $y_i$ is integral over $A$, hence $R$ is integral over $A$. The proof then states that the result follows, i.e that $R$ is a finitely generated $A$-module. I wasn't entirely sure how this follows from the above fact. My guess was that, if we let $y_i^{d_i}+\dots+c_i=0$ be a polynomial equation with coefficients in $A$ satisfied by $y_i$ for each $i\in\{1,\dots,n\}$, then we have $R=\sum_{i_1\in\{1,\dots,d_1\},\dots,i_n\in\{1,\dots,d_n\}}y_1^{i_1}\dots y_n^{i_n}A$
Is the above a generating set? If not could someone clarify how the result follows?
