Left and right ideals of $R=\left\{\bigl(\begin{smallmatrix}a&b\\0&c \end{smallmatrix}\bigr) : a\in\mathbb Z, \ b,c\in\mathbb Q\right\}$ If $$R=\left\{ \begin{pmatrix} a &b\\ 0 & c \end{pmatrix} \ : \ a \in \mathbb{Z}, \ b,c \in \mathbb{Q}\right\} $$
under usual addition and multiplication, then what are the left and right ideals of $R$?

This is covered in full on page 17 of Lam's First course in noncommutative rings. In general the "triangular ring" where $R$ and $S$ are rings and $M$ is an $R-S$ bimodule looks like: 
$$T= \begin{pmatrix} R &M\\ 0 & S \end{pmatrix} 
$$
You can also visualize the ring as $R\oplus M\oplus S$ with funny multiplication, but don't ever confuse this with ordinary direct sums. Lam explains:
1) The right ideals are all of the form $J_1\oplus J_2$, where $J_1$ is a right ideal of $R$ and $J_2$ is a right $S$ submodule of $M\oplus S$ which contains $J_1M$.
2) Analogously the left ideals are all of the form $I_1\oplus I_2$ where $I_2$ is a left ideal of $S$, and $I_1$ is a left $R$ submodule of $R\oplus M$ which contains $MI_2$.
3) The ideals of $T$ look like $K_1\oplus K_0\oplus K_2$ where $K_1$ is an ideal of $R$, $K_2$ is an ideal of $S$, and $K_0$ is a subbimodule of $M$ containing $K_1M+MK_2$.
As a bonus, believe I remember later somewhere he also shows that the radical of this ring is:
$$
rad(T)= \begin{pmatrix} rad(R) &M\\ 0 & rad(S) \end{pmatrix} 
$$
