In my Lie algebra course, the professor asked us to use Noether's Normalization Theorem and use it to say something about the Lie algebra of derivations over a commutative, associative algebra. Now, Noether Normalization theorem goes like this :
Let $A$ be a commutative finitely generated algebra over a field $k$. Then there exists algebraically independent elements $z_1, \dots, z_d\in A$ (i.e. $B=k[z_1, \dots, z_d]$ is isomorphic to a polynomial ring) such that $A$ is a finitely generated module over $B$.
Now the question posed by our professor is certainly open-ended, but I ask this question because what I'm thinking seems too strong and feels like it classifies a big class of derivation algebras, which seems unlikely. So far what I have got is that, $\text{Der}(k[x_1, \dots, x_n])$ can be thought of as a free module over $k[x_1, \dots, x_n]$ with generators given by $\frac{\partial}{\partial x_i}$. So for a finitely generated commutative algebra $A$, $\text{Der}(A)$ is the ring of derivations of $A$ which is a module over $k[x_1, \dots, x_n]$. Intuitively I want to say that $$\text{Der}(A)\cong \text{Der}(k[x_1, \dots, x_n])\otimes_{k[x_1, \dots, x_n]} A?$$ I know this is true when the module is a localization, i.e. if $A=k[x_1, \dots, x_n][f^{-1}]$ then my claim holds. I expect it fails generally, but does the Noether Normalization theorem make life simpler in any ways at all?
Any comments and suggestions welcomed, I don't want a full solution but maybe something that might suggest I am moving in the right direction or in the wrong direction. Thanks in advance.