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I'm sorry for asking such a simple question but I'm having a hard time thinking about all possible ideals of $M_2(Z)$.

So let's consider an arbitrary matrix $\begin{pmatrix} a & b \\ c & d\end{pmatrix}$ where $a,b,c,d \in Z$. How can I start constructing ideals?

user357335
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  • To begin, start thinking about the determinant – Ben Grossmann Nov 09 '20 at 01:09
  • What is so special about the determinant that lets me find ideals? Like, it's easy to think about ideals on $Z$ since it's easy to multiple all integers by something and analyze the result. But to multiple all matrices? I don't see a pattern to analyze. – user357335 Nov 09 '20 at 01:12
  • What I had in mind was the property $\det(AB) = \det(A) \det(B)$. Actually, I think I was steering you wrong there; I was thinking about subgroups of the multiplicative group over $M_2(F)$. – Ben Grossmann Nov 09 '20 at 01:16
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    At the very least, however, you can get ideals in $M_2(\Bbb Z)$ in the same way that you just described over $\Bbb Z$. For example, try to convince yourself that the matrices that contain only even entries form a ideal of $M_2(\Bbb Z)$. – Ben Grossmann Nov 09 '20 at 01:17
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    Two-sided ideals or left ideals? Also, are you familiar with the term "module"? It might help to think of $\Bbb Z^2$ as an $M_2(\Bbb Z)$-module, since when $M_2(\Bbb Z)$ acts on itself by left multiplication it's just $\Bbb Z^2\oplus\Bbb Z^2$ (a matrix is a pair of column vectors!). – anon Nov 09 '20 at 01:18

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