= Prime filtration
{title2=$M_i/M_{i-1}\cong R/P_i$}
A prime filtration of a <module> over a commutative <ring> is a finite increasing chain from zero to the module with each nonzero successive <quotient module> isomorphic to $R/P$ for a <prime ideal> $P$. Every <finitely generated module> over a <Noetherian ring> has one. In each nonzero quotient, a <maximal annihilator of a module element is prime> supplies a cyclic submodule $R/P$; lift it and repeat. The <ascending chain condition> for a <Noetherian module> makes the process terminate. This is not necessarily a composition series: the prime ideals need not be maximal.
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