Ideals of the subring $\begin{pmatrix} a & 0 \\ b & c \end{pmatrix}$ of $M_2(\mathbb Z)$

The set $A$ of all lower triangular $2\times 2$ matrices with entries in $\mathbb Z$ is a subring of $M_2(\mathbb Z)$. However, what are its ideals? I couldn't find them.

Suppose $I$ is an ideal of $A$. The elements of $I$, being also elements of $A$ have to be lower triangular and $I$ should be closed under left and right multiplication by lower triangular matrices. These facts couldn't lead me anywhere.

Edit: I saw the other question and the answer there went far above my head. I have no idea what a module is. Please help me do this without resorting to all of those advanced technique

• $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathcal{A} \iff b = 0$ – krirkrirk Jul 5 '17 at 14:46
• @krirkrirk yeah? – Cauchy Jul 5 '17 at 14:47
• What I meant is you need to check when $AB \in I$ for $B\in I$ and $A \in \mathcal{A}$, not for $A\in M_2(\mathbb{Z})$ – krirkrirk Jul 5 '17 at 14:50
• @krirkrirk just realized this. Thanks. I updated the question. – Cauchy Jul 5 '17 at 14:51
• rschwieb gives a good answer at the duplicate. I don't think it can be made easier. You do not need module theory to see the result. – Dietrich Burde Jul 5 '17 at 18:34