Editing Commutative ring (section)

=== Artinian rings ===
A ring is called [[Artinian ring|Artinian]] (after [[Emil Artin]]), if every descending chain of ideals
<math display="block"> R \supseteq I_0 \supseteq I_1 \supseteq \dots \supseteq I_n \supseteq I_{n+1} \dots </math>
becomes stationary eventually. Despite the two conditions appearing symmetric, Noetherian rings are much more general than Artinian rings. For example, <math> \mathbb{Z} </math> is Noetherian, since every ideal can be generated by one element, but is not Artinian, as the chain
<math display="block"> \mathbb{Z} \supsetneq 2\mathbb{Z} \supsetneq 4\mathbb{Z} \supsetneq 8\mathbb{Z} \dots </math>
shows. In fact, by the [[Hopkins–Levitzki theorem]], every Artinian ring is Noetherian. More precisely, Artinian rings can be characterized as the Noetherian rings whose Krull dimension is zero.