<s>[INST] Ideal in the ring of upper triangular matrices I'm trying to describe an ideal of the ring $R=\left\{ \begin{pmatrix}a & b\\
0 & c
\end{pmatrix}:a,b,c \in \mathbb{R}\right\} $
It's easy to prove that $I=\left\{ \begin{pmatrix}0 & a\\
0 & 0
\end{pmatrix}:a\in\mathbb{R}\right\} $ and $J=\left\{ \begin{pmatrix}a & b\\
0 & 0
\end{pmatrix}:a,b\in\mathbb{R}\right\} $ are ideals of $R$
My question is: how can I find other ideals?
Any help would be appreciated.
Thanks.
 [/INST] $\begin{pmatrix} a & b\\ 0 & c \end{pmatrix}
\begin{pmatrix} x & y\\ 0 & z \end{pmatrix} =
\begin{pmatrix} ax & ay+bz\\ 0 & cz \end{pmatrix}$.
If $a\neq 0$, you can vary $ax$ and $ay$ over $\mathbb{R}$, by varying $x$ and $y$. Similarly, if $b\neq 0$, you can vary $bz$ over $\mathbb{R}$, and if $c\neq 0$, you can vary $cz$ over $R$ by varying $z$. Thus, the different right ideals of $R$ are:
$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & \ast \end{pmatrix}, \begin{pmatrix} 0 & \ast \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & \ast \\ 0 & \ast \end{pmatrix}, \begin{pmatrix} \ast & \ast \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} \ast & \ast \\ 0 & \ast \end{pmatrix}$
</s>