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A module that is both Artinian and Noetherian has a finite composition series.
Artinian and Noetherian implies finite length: Let be an -module. If is both Artinian and Noetherian, then admits a finite composition series.
This combines Artinian and Noetherian chain conditions to produce a finite composition series, whose size is measured by module length.
Equivalently, has finite length.
An -module is Artinian if it satisfies the descending chain condition (DCC) on submodules: for every chain
there exists such that . For many classes of modules, Artinianity is equivalent to having finite length; compare length.
Artinian modules are “finite from below” in their submodule lattice and are the setting for induction on minimal submodules.
An -module is Noetherian if it satisfies the ascending chain condition (ACC) on submodules: for every chain
there exists such that .
Noetherian modules are the natural finiteness context for many arguments by maximality and stabilization.
A composition series of an -module is a finite chain of submodules
such that each factor is a simple module (a simple quotient in the sense of quotient modules). The integer is the length of the series and, when a composition series exists, is an invariant of (Jordan–Hölder), recorded as the module length.
Composition series exist exactly for modules of finite length and provide a canonical way to “factor” modules into simple pieces.
If an -module admits a composition series
then the length of , denoted , is the integer . By the Jordan–Hölder theorem, is well-defined (independent of the chosen composition series) whenever has at least one composition series; such modules are called “finite length.”
Finite length is tightly linked to chain conditions: modules that are both Noetherian and Artinian have finite length; see Artinian + Noetherian ⇒ finite length.