# Ideals of subrings of $M_2(\mathbb{Q})$

As a homework assignment, I've been given a particular subring of $M_2(\mathbb{Q})$, and asked to list all the ideals... For reference, $S$ = { set of matrices in $M_2(\mathbb{Q})$ with bottom left entry $0$ } is the subring in question.

I don't really see how to get a handle on this. Is there any sort of algorithmic way of doing this, or...? Where do I start?

One more thing is that ideals in your ring are also vector spaces over the Rationals, because the ring "contains" $\mathbb{Q}$ as the scalar matrices, so you know that once you have an ideal of dimension 2 it is maximal and cannot be extended (since the entire vector space is of dimension 3).