# Upper Triangular Matrix Rings and Noetherian/Artinian Hypotheses

## Left/Right Noetherian and Left/Right Artinian Rings

### Counterexamples in Upper Triangles Matrix Rings

The ring $\begin{bmatrix} \mathbb{Q} & \mathbb{Q} \\ 0 & \mathbb{Z} \end{bmatrix}$ is left Noetherian. However, it is not right Noetherian, left Artinian or right Artinian.

Similarly, the ring $\begin{bmatrix} \mathbb{R} & \mathbb{R} \\ 0 & \mathbb{Q} \end{bmatrix}$ is left Artinian and left Noetherian, but not right Artinian or right Noetherian.

The details are worked out here under “CIA: Some Upper Triangular Matrix Rings and Noetherian/Artinian Hypotheses.”