Editing Artinian module (section)

==Relation to the Noetherian condition==
Unlike the case of rings, there are Artinian modules which are not [[Noetherian module]]s. For example, consider the ''p''-primary component of <math>\mathbb{Q}/\mathbb{Z}</math>, that is <math>\mathbb{Z}[1/p] / \mathbb{Z}</math>, which is [[isomorphic]] to the ''p''-[[quasicyclic group]] <math>\mathbb{Z}(p^\infty)</math>, regarded as <math>\mathbb{Z}</math>-module. The chain <math>\langle 1/p \rangle \subset \langle 1/p^2 \rangle \subset \langle 1/p^3 \rangle \subset \cdots</math> does not terminate, so <math>\mathbb{Z}(p^\infty)</math> (and therefore <math>\mathbb{Q}/\mathbb{Z}</math>) is not Noetherian. Yet every descending chain of submodules terminates, since any proper submodule has the form <math>\langle 1/n \rangle</math>  for some integer <math>n</math> and is therefore a finite set; so <math>\mathbb{Z}(p^\infty)</math> is Artinian.

Note that <math>\mathbb{Z}[1/p] / \mathbb{Z}</math> is also a [[faithful module|faithful]] <math>\mathbb{Z}</math>-module. So, this also provides an example of a faithful Artinian module over a non-Artinian ring. This does not happen for Noetherian case; if ''M'' is a faithful Noetherian module over ''A'' then ''A''  is Noetherian as well.

Over a [[commutative ring]], every [[cyclic module|cyclic]] Artinian module is also Noetherian, but over noncommutative rings cyclic Artinian modules can have uncountable [[length of a module|length]] as shown in the article of Hartley and summarized in the [[Paul Cohn]] article dedicated to Hartley's memory.

Another relevant result is the [[Akizuki–Hopkins–Levitzki theorem]], which states that the Artinian and Noetherian conditions are equivalent for modules over a [[semiprimary ring]].