Commutative algebra – When does a set of polynomials form a polynomial ring?

I think about 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 join all the elements of $ S $ at k $ a polynomial ring on the set $ S $?

I made the following observations:

  • of course we should have that $ S $ contains $ n $ elements or less, otherwise we can find relationships.

  • 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, I guess the answer is that $ S = {s_1, … } $ must be such that $ s_i $ is not already an element in k $[s_1,…,s_{i-1}]$. Are there any "easier" criteria for this? For some ideal $ J subset k[x_1,…,x_n]$, can we still extract a generator $ S = {a_1, …, a_c } $ such as k $[a_1,…,a_c]$ is a polynomial ring? Under what assumptions is it possible, if it is not possible in general?