Noetherian
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).
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.
(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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