## Definition ## Given a [[commutative ring]] $R$, a **filtered $R$-algebra** is an [[algebra (ring theory)|$R$-algebra]] $A$ whose underlying abelian group is a [[graded module|$\mathbb{N}$-graded $R$-module]], such that for natural numbers $m:\mathbb{N}$ and $n:\mathbb{N}$, the product of every $m$-multivector and $n$-multivector is an $m+n$-multivector: $$\prod_{m:\mathbb{N}} \prod_{n:\mathbb{N}} \prod_{a:\mathcal{F}_m(A)} \prod_{b:\mathcal{F}_n(A)} \left[\sum_{c:\mathcal{F}_{m+n}(A)} a \cdot b = c\right]$$ Every [[geometric algebra|geometric $R$-algebra]] is a filtered $R$-algebra. ## See also * [[graded module]] * [[geometric algebra]]