= Filtered algebra
{wiki}
A <filtered algebra> over a <field> $k$ has increasing vector subspaces $F_iA$ with $F_iA\,F_jA\subseteq F_{i+j}A$. In the usual exhaustive nonnegative convention, $F_{-1}A=0$, $1\in F_0A$, and $\bigcup_{i\geq0}F_iA=A$. Multiplication on the <associated graded ring> is induced by multiplying representatives and retaining their highest filtered degree.
For a <polynomial ring>, polynomials of degree at most $i$ give this filtration. For a <Weyl algebra>, the <Bernstein filtration> assigns degree one to coordinates and derivatives. Its defining commutators have lower degree, so the associated graded algebra becomes commutative. A <filtration of a module> compatible with the algebra filtration similarly gives an associated graded module.
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