Artinian module

Artinian module

In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its submodules. They are for modules what Artinian rings 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 for Emil 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 modules. 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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