View source for 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 sequence|composition series]] if and only if it is both Artinian and [[Noetherian module|Noetherian]]. See also [[Artinian ring]].


====References====
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|valign="top"|{{Ref|Fa}}||valign="top"|  C. Faith,   "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)
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|valign="top"|{{Ref|Fa2}}||valign="top"|  C. Faith,   "Algebra" , '''II. Ring theory''' , Springer  (1976)
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[[Category:Associative rings and algebras]]