Difference between revisions of "Noetherian"
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We say that a ring <math>R</math> is left (right) noetherian if it is noetherian as a left (right) <math>R</math>-module. If <math>R</math> is both left and right noetherian, we call it simply noetherian. | We say that a ring <math>R</math> is left (right) noetherian if it is noetherian as a left (right) <math>R</math>-module. If <math>R</math> is both left and right noetherian, we call it simply noetherian. | ||
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| + | [[Category:Ring theory]] | ||
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Revision as of 18:03, 9 September 2008
Let 
be a ring and 
a left -module. Then we say that is a noetherian module if it satisfies the following property, known as the ascending chain condition (ACC):
- For any ascending chain
of submodules of 
, there exists an integer 
so that 
(i.e. the chain eventually terminates).
of 
so that 
(i.e. the chain eventually terminates).Theorem. The following conditions are equivalent for a left -module:
- is noetherian.
- Every submodule
of is finitely generated (i.e. can be written as 
for some 
). - For every collection of submodules of , there is a maximal element.
of 
for some 
).(The second condition is also frequently used as the definition for noetherian.)
We also have right noetherian modules with the appropriate adjustments.
We say that a ring is left (right) noetherian if it is noetherian as a left (right) -module. If is both left and right noetherian, we call it simply noetherian.
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