Artinian module
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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.
[edit] 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.
[edit] 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 
, that is ![\mathbb{Z}[1/p] / \mathbb{Z}](../../../../math/d/4/d/d4d1b33dc3f98021dc6ece8b2ef9f71f.png)
, which is isomorphic to the p-quasicyclic group 
, regarded as 
-module. The chain 
does not terminate, so (and therefore ) is not Noetherian. Yet every descending chain of (without loss of generality proper) submodules terminates: Each such chain has the form 
for some integers n1,n2,n3, ..., and the inclusion of 
implies that ni + 1 must divide ni. So n1,n2,n3, ... is a decreasing sequence of positive integers. Thus the sequence terminates, making 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 Cohn article dedicated to Hartley's memory.
[edit] References
- Atiyah, M.F.; Macdonald, I.G. (1969). "Chapter 6. Chain conditions; Chapter 8. Artin rings", Introduction to Commutative Algebra. Westview Press. ISBN 978-0201407518.
- Cohn, P.M. (1997). "Cyclic Artinian Modules Without a Composition Series". J. London Math. Soc. (2) 55 (2): 231—235. doi:.
- Hartley, B. (1977). "Uncountable Artinian modules and uncountable soluble groups satisfying Min-n". Proc. London Math. Soc. (3) 35 (1): 55—75.
- Lam, T.Y. (2001). "Chapter 1. Wedderburn-Artin theory", A First Course in Noncommutative Rings. Springer Verlag. ISBN 978-0387953250.
