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