**Yoshifumi Tsuchimoto**

- If is in the Jacobson radical of , then for any finite -module , we have . Furthermore, for any submodule of , we have .
- If is an integral domain and is a proper ideal of , then we have .

- (which is also equivalent to saying that or that ).
- .
- Any descending chain

- .
- For any ideal of definition of , The leading coefficient of coincides with that of .