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.”

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s