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?

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

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2019-12-02 02:53:19
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Answers: 1

Start by considering perfects created by one component - pick a matrix and also see what takes place when you increase it from the left and also right, etc

Every perfect has an excellent created by one component, so currently you require to see if you can expand several of the perfects you reached bigger perfects.

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).

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2019-12-03 04:20:16
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