I am currently looking at the following example (and other similar examples) and I can follow the proof that it is a right Artinian ring and I also follow the example given as to why it is not a left Artinian ring.
However, I do not understand what is stopping us from applying the reasoning used to show it is right Artinian to conclude that it is also left Artinian. I do not see which bit of the proof fails and was wondering if anyone could help? I have also attached the statements of the theorems used in the proof.
The second sentence in the second paragraph fails to be true when you consider the left module structure. As a right $R$-module, $J$ does not contain any submodules except for $0$ and $J$. However, as a left $R$-module $J$ contains an infinite descending chain of submodules.
To see this, note that the right $R$-module structure on $J$ is essentially the $\mathbb Q(x)$-module structure on $\mathbb Q(x)$. However, if you multiply $J$ with $R$ from the left, you get the $\mathbb Q$-module structure on $\mathbb Q(x)$. Now, as $\mathbb Q(x)$-module, $\mathbb Q(x)$ is a $1$-dimensional vector space, i.e., simple. However as a $\mathbb Q$-module, $\mathbb Q(x)$ is infinite-dimensional, hence not Artinian.
The same reasoning works, for instance, for the ring $\begin{pmatrix} \mathbb Z & \mathbb Q \\ 0 & \mathbb Q \end{pmatrix}$: ${}_{\mathbb Z} \mathbb Q$ is not Artinian, but $\mathbb Q_{\mathbb Q}$ is.