## Defintion ## Given a [[commutative ring]] $R$, a **geometric $R$-algebra** is a [[filtered algebra|filtered $R$-algebra]] $A$ with a [[ring isomorphism]] $j:\langle A \rangle_0 \cong R$ such that the product of every $1$-vector with itself is a $0$-vector. $$\prod_{a:\langle A \rangle_1} \left[\sum_{c:\langle A \rangle_0} a \cdot a = c\right]$$ The $0$-vectors are called **scalars** and $1$-vectors are just called **vectors** Every geometric $R$-algebra is a $R$-[[Clifford algebra]]. ## See also ## * [[Clifford algebra]] * [[filtered algebra]] * [[real geometric algebra]] * [[graded module]] ## References ## * G. Aragón, J.L. Aragón, M.A. Rodríguez (1997), Clifford Algebras and Geometric Algebra, _Advances in Applied Clifford Algebras_ Vol. 7 No. 2, pg 91–102, doi:10.1007/BF03041220, S2CID:120860757