My problem involves considering the ideal $I = \{ X^mY^n \mid m,n\in \mathbb{N}, m^2n>5 \}$ of $\mathbb{Q}[X, Y]$. I am asked to write down a finite generating set of $I$ and explain how I arrived at the answer.
I can see that if there weren't the restriction that $m^2n$ is greater than $5$ then $XY$ would generate the ideal but I don't know how to incorporate that restriction into the finite generating set. Any clues would be appreciated!
Draw the grid $\mathbb{N}^2$ in the first quadrant, where the point $(m,n)$ corresponds to the generator $X^mY^n$. Your ideal is generated by all the coordinate points strictly above the curve $m^2 n = 5$ (or $n = 5/m^2$).
Note that if a point $p = (m,n)$ is in your ideal, then it also generates all the points above and to the right of $p$. So, look at the picture and figure out a finite set of points $p_i$, such that the union of the regions above and to the right of each $p_i$ gives all of $I$.
(Note: $X^m$ and $Y^n$ are never in the ideal, so start with $m=1$ to get $X Y^6$, and $n=1$ to get $X^3 Y$.)
Picture: