For example, Z is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain Z ⊋ 2Z ⊋ 4Z ⊋ 8Z ⊋ ... shows.
例如，Z 是诺特的，因为任何理想可由一个元素生成，但不是阿廷环，比如有降链 Z ⊋ 2Z ⊋ 4Z ⊋ 8Z ⊋ ...。