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Given a commutative ring , a -pre-Clifford algebra is an -algebra with a canonical injection and a quadratic form such that
A -pre-Clifford algebra homomorphism between two -pre-Clifford algebras and is an -algebra homomorphism such that the quadratic form is preserved
The -Clifford algebra is the initial object in the category of -pre-Clifford algebras and -pre-Clifford algebra homomorphisms.
Suppose an -dimensional -Clifford algebra has a basis vector function , where
is the type of natural numbers less than , such that for all , . Then is the standard diagonal form and is usually written as .