Difference between revisions of "Artinian"
(New page: We say that a ring or module is '''Artinian''' if the descending chain condition holds for its ideals/submodules. The notion is similar to that of Noetherian rings...) |
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We say that a [[ring]] or [[module]] is '''Artinian''' | We say that a [[ring]] or [[module]] is '''Artinian''' | ||
if the [[descending chain condition]] holds for its | if the [[descending chain condition]] holds for its | ||
| − | [[ideals]]/[[submodules]]. The notion is similar to that | + | [[ideal | ideals]]/[[submodules]]. The notion is similar to that |
of [[Noetherian]] rings and modules. | of [[Noetherian]] rings and modules. | ||
One might expect Artinian rings to be just as broad and | One might expect Artinian rings to be just as broad and | ||
diverse a category as Noetherian rings. However, this | diverse a category as Noetherian rings. However, this | ||
| − | is not the case. | + | is not the case. In fact, Artinian rings are Noetherian but the converse does not hold. |
| + | '''Theorem.''' Let <math>A</math> be an Artinian ring. Then every prime ideal of <math>A</math> is maximal. Thus <math>\dim A = 0</math> (the [[Krull dimension]] of <math>A</math> is <math>0</math>). | ||
'''Theorem.''' Let <math>A</math> be a ring. Then <math>A</math> is Artinian | '''Theorem.''' Let <math>A</math> be a ring. Then <math>A</math> is Artinian | ||
if and only if <math>A</math> is Noetherian and every element of <math>A</math> | if and only if <math>A</math> is Noetherian and every element of <math>A</math> | ||
is either [[invertible]] or [[nilpotent]]. | is either [[invertible]] or [[nilpotent]]. | ||
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| − | |||
| − | |||
However, Artinian ''modules'' are not necessarily | However, Artinian ''modules'' are not necessarily | ||
Latest revision as of 20:00, 19 April 2012
We say that a ring or module is Artinian if the descending chain condition holds for its ideals/submodules. The notion is similar to that of Noetherian rings and modules.
One might expect Artinian rings to be just as broad and diverse a category as Noetherian rings. However, this is not the case. In fact, Artinian rings are Noetherian but the converse does not hold.
Theorem. Let 
be an Artinian ring. Then every prime ideal of is maximal. Thus 
(the Krull dimension of is 
).
Theorem. Let be a ring. Then is Artinian
if and only if is Noetherian and every element of
is either invertible or nilpotent.
However, Artinian modules are not necessarily
Noetherian. Consider, for example, the Prüfer Group
for some prime 
as a 
-module (i.e.,
the additive group of rationals of the form 
, modulo
). Each of its submodules is of the form
, for some integer 
. Thus a descending
chain of submodules corresponds uniquely to an increasing
sequence of nonnegative integers, and vice-versa.
Thus every ascending chain must stabilize, but we have
the descending chain
![\[(1/p^0) \supset (1/p^1) \supset (1/p^2) \supset \dotsb .\]](http://latex.artofproblemsolving.com/4/9/a/49ab11d2d2a499f3e10b2b3e30eeb93627e69295.png)
This module is therefore Artinian, but not Noetherian.
