# Subring of polynomial ring containing two polynomials of relatively prime degree

Let $$R$$ be an integral domain , $$S$$ be a subring of $$R[x_1,..., x_n]$$ .

If $$S$$ contains two polynomials of relatively prime degree, then how to show that there exists an integer $$m>1$$ such that $$S$$ contains a polynomial of degree $$l$$ for every $$l>m$$ ?

I can easily show the claim for $$n=1$$, but having difficulty in showing for higher $$n$$. Please help.

• How did you do it for $n=1$? – Eric Wofsey Mar 6 at 22:58

Hint: Prove that if a subset $$A\subset\Bbb{N}$$ is closed under taking sums, and if $$A$$ contains two relatively prime elements, then there exists an integer $$m>1$$ such that $$A$$ contains all natural numbers $$l>m$$.