Is $\langle X^2Y + Y^2X \rangle$ a monomial ideal in $K[X,Y]$?

Question

Is $\langle X^2Y + Y^2X \rangle$ a monomial ideal in $K[X,Y]$ (where $K$ is a field)?

I have a feeling the answer is no, but I am having trouble justifying it.

Can anyone show me how to prove this?

I suppose that if this is indeed a monomial ideal, then every monomial in $X^2Y + Y^2X $ must be divisible by a monomial in our generating set. So every monomial in our generating set must be of degree $\leqslant 2$. Is this fact helpful in proving it?

Answer 1

No. If it were, then $X^2Y$ and $XY^2$, being monomials within the generator $X^2Y+XY^2$, would be in the ideal. They are not.