Left/Right Noetherian and Left/Right Artinian Rings
Counterexamples in Upper Triangles Matrix Rings
The ring is left Noetherian. However, it is not right Noetherian, left Artinian or right Artinian.
Similarly, the ring 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.”