Left- and right artinian

Question

Let $R$ be the ring defined by $$R=\left\{\begin{pmatrix}a&0\\c&d\end{pmatrix}\mid a\in\mathbb{Q},c,d\in\mathbb{R}\right\}.$$ I want to show that it is left-artinian. I have calculated all possible left ideals: $$L_1=\begin{pmatrix}0&0\\0&0\end{pmatrix} $$ $$L_2=\begin{pmatrix}\mathbb{Q}&0\\\mathbb{R}&\mathbb{R}\end{pmatrix}$$ $$L_3=\begin{pmatrix}0&0\\\mathbb{R}&0\end{pmatrix} $$ $$L_4=\begin{pmatrix}0&0\\0&\mathbb{R}\end{pmatrix} $$ $$L_5=\begin{pmatrix}\mathbb{Q}&0\\\mathbb{R}&0\end{pmatrix} $$ $$L_6=\begin{pmatrix}0&0\\\mathbb{R}&\mathbb{R}\end{pmatrix} $$ $$L_7=\mathbb{R}\begin{pmatrix}0&0\\1&r\end{pmatrix} $$for $r \in\mathbb{R}.$ How can I conclude that $R$ is left artinian?