# 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 $$S$$?

• 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?