noetherian

A module $M$ is noetherian if it satisfies the following equivalent conditions:

•
the ascending chain condition holds for submodules of $M$ ;
•
every nonempty family of submodules of $M$ has a maximal element;
•
every submodule of $M$ is finitely generated.

A ring $R$ is left noetherianif it is noetherian as a left module over itself(i.e. if ${}_{R}R$ is a ),and right noetherianif it is noetherian as a right module over itself(i.e. if $R_{R}$ is an ),and simply noetherianif both conditions hold.

单词	Noetherian
释义	noetherian A module $M$ is noetherian if it satisfies the following equivalent conditions: • the ascending chain condition holds for submodules of $M$ ; • every nonempty family of submodules of $M$ has a maximal element; • every submodule of $M$ is finitely generated. A ring $R$ is left noetherianif it is noetherian as a left module over itself(i.e. if ${}_{R}R$ is a ),and right noetherianif it is noetherian as a right module over itself(i.e. if $R_{R}$ is an ),and simply noetherianif both conditions hold.
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