Example of non-graded ideal

Question

Let $R=k[x,y]$ and $M=(x^2+y^3)$, and take the usual grading on $R$ (i.e. $R_n$ are the homogeneous polynomials of degree $n$). We have that $M$ is not a graded submodule. This is because if we define $M_n=M\cap R_n$, we have that $M_0=M_1=M_2=0$, so the map from the direct sum of the $M_n$'s to $M$ cannot be surjective. This is something my lecturer wrote on the board and I agree with him. However, I think that more can be said. I think that we can assure that all $M_n$'s are $0$. This is because the polynomial that generates $M$ is not homogeneous, so no element of the form $p(x^2+y^3)$ can be contained in a subgroup $R_n$ for some $n$, where $p$ denotes an element of $k[x,y]$. Am I right in asserting this, or is there a mistake in my reasoning?