- Artinian module
In

abstract algebra , an**Artinian module**is a module that satisfies thedescending chain condition on its submodules. They are for modules whatArtinian ring s are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named forEmil Artin .Like

Noetherian modules , Artinian modules enjoy the following heredity property:

* If "M" is an Artinian "R"-module, then so is any submodule and any quotient of "M".The converse also holds:

* If "M" is any "R" module and "N" any Artinian submodule such that "M"/"N" is Artinian, then "M" is Artinian.As a consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also Noetherian, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring "R", any finitely-generated "R"-module is both Noetherian and Artinian, and is said to be of finite length; however, if "R" is not Artinian, or if "M" is not finitely generated, there are counterexamples.**Left and right Artinian modules**If the ring of definition is "R", then as with the condition that "R" itself be Artinian, when "R" is not commutative there is some distinction between the concepts of left- and right-Artinian modules over "R". Namely, "R" is said to be left Artinian if, as a module over itself via multiplication on the left, it is Artinian; likewise right Artinian. However, if "M" is any left "R"-module which is Artinian, then it is by definition left Artinian and the distinction need not be made. Occasionally the same abelian group "M" is realized as both a left and a right "R"-module in different ways, in which case, to separate the properties of the two structures, one can abuse notation and refer to "M" as left Artinian or right Artinian when, strictly speaking, it is correct to say that "M", with its left "R"-module structure, is Artinian, etc.

**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 $mathbb\{Q\}/mathbb\{Z\}$, that is $mathbb\{Z\}\; [1/p]\; /\; mathbb\{Z\}$, which is isomorphic to the "p"-quasicyclic group $mathbb\{Z\}(p^\{infty\})$, regarded as $mathbb\{Z\}$-module. The chain $langle\; 1/p\; angle\; subset\; langle\; 1/p^2\; angle\; subset\; langle\; 1/p^3\; angle\; cdots$ does not terminate, so $mathbb\{Z\}(p^\{infty\})$ (and therefore $mathbb\{Q\}/mathbb\{Z\}$) is not Noetherian. Yet every descending chain of (without loss of generality proper) submodules terminates: Each such chain has the form $langle\; 1/n\_1\; angle\; supseteq\; langle\; 1/n\_2\; angle\; supseteq\; langle\; 1/n\_3\; angle\; cdots$ for some integers $n\_1\; ,\; n\_2\; ,\; n\_3,$ ..., and the inclusion of $langle\; 1/n\_\{i+1\}\; angle\; subseteq\; langle\; 1/n\_i\; angle$ implies that $n\_\{i+1\}$ must divide $n\_i$. So $n\_1\; ,\; n\_2\; ,\; n\_3,$ ... is a decreasing sequence of positive integers. Thus the sequence terminates, making $mathbb\{Z\}(p^\{infty\})$ Artinian.Over a commutative ring, every cyclic Artinian module is also Noetherian, but over noncommutative rings cyclic Artinian modules can have uncountable length as shown in the article of Hartley and summarized nicely in the

Paul Cohn article dedicated to Hartley's memory.**References*** cite book

last = Atiyah

first = M.F.

authorlink = Michael Atiyah

coauthors = Macdonald, I.G.

title = Introduction to Commutative Algebra

isbn = 978-0201407518

year = 1969

publisher = Westview Press

chapter = Chapter 6. Chain conditions; Chapter 8. Artin rings

* cite journal

last = Cohn

first = P.M.

authorlink = Paul Cohn

title = Cyclic Artinian Modules Without a Composition Series

journal = J. London Math. Soc. (2)

volume = 55

issue = 2

pages = 231–235

year = 1997

url = http://www.ams.org/mathscinet-getitem?mr=442091

doi = 10.1112/S0024610797004912

* cite journal

last = Hartley

first = B.

title = Uncountable Artinian modules and uncountable soluble groups satisfying Min-n

journal = Proc. London Math. Soc. (3)

volume = 35

issue = 1

pages = 55–75

year = 1977

url = http://www.ams.org/mathscinet-getitem?mr=442091

doi = 10.1112/plms/s3-35.1.55

* cite book

last = Lam

first = T.Y.

title = A First Course in Noncommutative Rings

isbn = 978-0387953250

year = 2001

publisher = Springer Verlag

chapter = Chapter 1. Wedderburn-Artin theory

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