I am trying to solve an exercise about Artinian an Noetherian rings of $2 \times 2$ matrices but I really can't get to a solution. The exercise is the following: Set $$ R = \left\{ \begin{pmatrix} q & 0 \\ r & s \\ \end{pmatrix} \mid q \in \mathbb{Q}, \ r,s \in \mathbb{R} \right\}. $$ Show that $R$ is left artinian and left noetherian, but it is neither right artinian nor right noetherian.
I think I managed to prove it is left noetherian (showing that every left ideal is finitely generated), but I can't find a way to prove it is artinian since (I guess) it is necessary to show explicitly that every descending chain of left submodules is stationary.
Could you please help me?
Let $V$ be a $\Bbb Q$-vector subspace of $\Bbb R$. Then the set of $$\pmatrix{0&0\\v&0}$$ with $v\in V$ is a right ideal of $R$. As $\Bbb R$ is an infinite-dimensional $\Bbb Q$-vector space, it is easy to prove that it has non-trivial ascending and descending sequences of $\Bbb Q$-vector subspaces.