= Ascending filtration
{title2=$F_{-1}=0\subseteq F_0\subseteq F_1\subseteq\cdots$}
An ascending filtration is an increasing chain of additive subgroups, or vector subspaces when a scalar field is fixed. It is exhaustive if their union is the whole object. For a <filtered ring>, multiplicative compatibility means $F_iR\,F_jR\subseteq F_{i+j}R$. For an ideal $I$, intersecting each level with $I$ gives the induced filtration. Its <associated graded module> has components $(I\cap F_nR)/(I\cap F_{n-1}R)$.
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