# 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?

Edit: I posted this over at Ask an Algebraist originally: http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_algebraist;task=show_msg;msg=2257

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## 1 Answer

Start by looking at ideals generated by one element - choose a matrix and see what happens when you multiply it from the left and right, etc.

Every ideal contains an ideal generated by one element, so now you need to see if you can extend some of the ideals you got to larger ideals.

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).

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