Noetherian module: Difference between revisions
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==Definition== | ==Definition== | ||
Revision as of 00:02, 8 January 2008
This article defines a property of a module over a commutative unital ring
A commutative unital ring which satisfies this property as a module over itself, is termed a/an: Noetherian ring
Definition
Symbol-free definition
A module over a commutative unital ring is termed Noetherian if every submodule of it is finitely generated.
Facts
- A ring is Noetherian as a module over itself if and only if it is a Noetherian ring. This is because ideals of a ring are precisely its submodules.
- Every finitely generated module over a Noetherian ring is a Noetherian module.