A module M is Noetherian if it obeys the ascending chain condition with respect to inclusion, i.e., if every set of increasing sequences of submodules eventually becomes constant. If a module M is Noetherian, then the following are equivalent. 1.M satisfies the ascending chain condition on submodules. 2. Every submodule of M is finitely generated. 3. Every set of submodules of M contains a maximal element.
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