Difference between revisions of "Artinian module"
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| − | A [[module]] that satisfies the descending [[chain condition]] for [[submodule]]s. The class of Artinian modules is closed with respect to passing to submodules, quotient modules, finite [[direct sum]]s and [[Extension of a module|extension]]s. Extension in this context means that if the modules $B$ and $A/B$ are Artinian, then so is $A$. Each Artinian module can be decomposed into a direct sum of submodules which are no longer decomposable into a direct sum. A module has a composition series if and only if it is both Artinian and [[Noetherian module|Noetherian]]. See also [[Artinian ring]]. | + | A [[module]] that satisfies the descending [[chain condition]] for [[submodule]]s. The class of Artinian modules is closed with respect to passing to submodules, quotient modules, finite [[direct sum]]s and [[Extension of a module|extension]]s. Extension in this context means that if the modules $B$ and $A/B$ are Artinian, then so is $A$. Each Artinian module can be decomposed into a direct sum of submodules which are no longer decomposable into a direct sum. A module has a [[Composition sequence|composition series]] if and only if it is both Artinian and [[Noetherian module|Noetherian]]. See also [[Artinian ring]]. |
Latest revision as of 19:06, 25 October 2014
A module that satisfies the descending chain condition for submodules. The class of Artinian modules is closed with respect to passing to submodules, quotient modules, finite direct sums and extensions. Extension in this context means that if the modules $B$ and $A/B$ are Artinian, then so is $A$. Each Artinian module can be decomposed into a direct sum of submodules which are no longer decomposable into a direct sum. A module has a composition series if and only if it is both Artinian and Noetherian. See also Artinian ring.
References
| [Fa] | C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) |
| [Fa2] | C. Faith, "Algebra" , II. Ring theory , Springer (1976) |
How to Cite This Entry:
Artinian module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artinian_module&oldid=34019
Artinian module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artinian_module&oldid=34019
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article