## Defintion ## Given a [[commutative ring]] $R$, a $R$-**pre-Clifford algebra** is an $R$-[[algebra (ring theory)|algebra]] $A$ with a canonical injection $\iota: R \to A$ and a [[quadratic form]] $q:A \to R$ such that $$\prod_{a:A} a \cdot a = \iota(q(a))$$ A $R$-**pre-Clifford algebra homomorphism** between two $R$-pre-Clifford algebras $A$ and $B$ is an $R$-algebra homomorphism $f:A \to B$ such that the quadratic form is preserved $$\prod_{a:A} q_A(a) = q_B(f(a))$$ The $R$-**Clifford algebra** $Cl(R, q)$ is the initial object in the category of $R$-pre-Clifford algebras and $R$-pre-Clifford algebra homomorphisms. Suppose an $n$-dimensional $R$-Clifford algebra $A$ has a basis vector function $e:U(n) \to A$, where $$U(n) \coloneqq \sum_{i:\mathbb{N}} i \lt n$$ is the type of natural numbers less than $n$, such that for all $i:U(n)$, $q(e_i) = 1$. Then $q$ is the standard diagonal form and $A$ is usually written as $Cl_n(R)$. ## See also ## * [[algebra (ring theory)]] * [[geometric algebra]]