Monomials not in an ideal Let $R=\mathbb{R}[x,y]$ denote the commutative ring of polynomials in two variables $x,y$ with real coefficients. Show that for each $k \in \mathbb{N}$ there exists a monomial of degree $k$ not belonging to the ideal generated in $R$ by $p(x,y)=x^2+y^2$. 
P.S. I know the elements of the ideal are of the form $q(x,y)(x^2+y^2)$ for $q(x,y)\in R$. I'm trying to prove, for example, that $x^k$ is not of that form. I'm asking for the reason why... (it must be some obvious thing i cant see)

Suppose $x^k = q(x,y)(x^2+y^2)$. 
$q$ has no $y^n$ term, otherwise the product would have $y^{n+2}$. Similarly, $q$ has no $x^ny^m$ term. So $q$ is a polynomial in $x$, but this too is impossible. Therefore, $x^k$ is not in the ideal.
In fact, you can convince yourself in a similar manner that all nonzero elements of the ideal are at least binomial.
