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.