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 noetherian if it is noetherian as a left module over itself (i.e. if $_RR$ is a noetherian module), and right noetherian if it is noetherian as a right module over itself (i.e. if $R_R$ is an noetherian module), and simply noetherian if both conditions hold.