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  "body": "\n## Definition\n\nLet $K$ be a [[commutative ring|0GYGHKX834]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $K$ with unit $1$.\nAn *(increasing, exhaustive) filtration* of $A$ is a family $(F_i)_{i \\ge 0}$ of $K$-submodules of $A$ such that:\n\n1. $F_i \\subseteq F_{i+1}$ for all $i \\ge 0$.\n2. $\\bigcup_{i \\ge 0} F_i = A$.\n3. $F_i F_j \\subseteq F_{i+j}$ for all $i, j \\ge 0$.\n4. $1 \\in F_0$.\n\nA *filtered algebra* is an algebra together with such a filtration.\n\n## Remarks\n\nThe *associated graded algebra* is $\\operatorname{gr} A = \\bigoplus_{i \\ge 0} F_i / F_{i-1}$ with $F_{-1} = 0$; clause 3 makes the product $(a + F_{i-1})(b + F_{j-1}) = ab + F_{i+j-1}$ well defined.\nExample: $A = K[x]$ with $F_i$ the polynomials of degree at most $i$.\nClauses 1, 2 and 4 are clear, and clause 3 holds since $\\deg(fg) \\le \\deg f + \\deg g$.\nHere $\\operatorname{gr} A \\cong K[x]$.\nNon-example: in $K[x]$ take $F_0 = K + Kx$ and $F_i$ the polynomials of degree at most $i + 1$ for $i \\ge 1$.\nClauses 1, 2 and 4 hold, but clause 3 fails: $x \\cdot x = x^2 \\notin F_0$.\nLet $U(\\mathfrak{g})$ be the universal enveloping algebra of a Lie algebra $\\mathfrak{g}$ over a field, with $F_i$ spanned by products of at most $i$ elements of $\\mathfrak{g}$.\nThen the Poincaré–Birkhoff–Witt theorem gives $\\operatorname{gr} U(\\mathfrak{g}) \\cong S(\\mathfrak{g})$.\n",
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