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

As a research assignment, I've been offered a certain subring of $M_2(\mathbb{Q})$, and also asked to detail all the perfects ... For reference, $S$ = set of matrices in $M_2(\mathbb{Q})$ with lower left access $0$ is the subring concerned.

I do not actually see just how to handle this. Exists any kind of type of mathematical means of doing this, or ...? Where do I start?

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2019-12-02 02:53:19
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Another point is that perfects in your ring are additionally vector rooms over the Rationals, due to the fact that the ring "contains" $\mathbb{Q}$ as the scalar matrices, so you recognize that as soon as you have a perfect of measurement 2 it is topmost and also can not be expanded (given that the whole vector room is of measurement 3).