I am considering a problem in Ring Theory.
Consider the ring $\mathbb{M}_{2}(\mathbb{Z})$,the 2x2 matrices with entries in $\mathbb{Z}$. Find all its ideals.
So the way I start thinking this problem is, I first try think about $\mathbb{M}_{2}(\mathbb{R})$. By elementary matrix's row reduction and column reduction, I found that in any non-trivial ideal in $\mathbb{M}_{2}(\mathbb{R})$ it can be shown to contain the identity $\mathbf{I}$ thus the ideal is $\mathbb{M}_{2}(\mathbb{R})$ itself.
Then I turn to think about, what happens if I restrict the entry from $\mathbb{R}$ to $\mathbb{Z}$. The observation I make is that, we cannot apply the trick as in $\mathbb{M}_{2}(\mathbb{R})$ to obtain an identity $\mathbf{I}$ because we don't necessarily have multiplicative inverse for $\mathbb{Z}$, like 2's inverse is $\frac{1}{2} \notin \mathbb{Z}$.
So from here What I conclude is that, since we're not guarantee to have identity $\mathbf{I}$ in ideal, $\mathbb{M}_{2}(\mathbb{Z})$ should have some proper ideal. But I am stuck on how to find all such ideal. It's easy to think for a few, like upper-triangular matrix, lower-triangular matrix. But how can I show them all, or prove there is no other.
Thanks in advance. Any hint strongly appreciated~!
