I am pondering over the following question: Suppose $k$ is a field and we are given a set $S\subseteq k[x_i,...,x_n]$ of polynomials. When is the ring $k[S]$ where we adjoin all the elements of $S$ to $k$ a polynomial ring on the set $S$?
I have made the following observations:
of course one should have that $S$ contains $n$ elements or less, otherwise we can find some relations.
something like $S=\{x^2,x^3\}\subset k[x]$ does not work because $(x^2)^3=(x^3)^2$.
- something like $S=\{x,y,xy\}\subset k[x,y,z]$ does not work because xy=x*y.
So my guess is, that the answer is that $S=\{s_1,...\}$ has to be such that $s_i$ is not already an element in $k[s_1,...,s_{i-1}]$. Are there ''easier" criterions for this? For some ideal $J\subset k[x_1,...,x_n]$, can one always extract a generating set $S=\{a_1,...,a_c\}$ such that $k[a_1,...,a_c]$ is a polynomial ring? Under which assumptions is this possible, if this is not possible in general?