The dual module \(M^* : M \to _{l[R]} R\) is the \(R\)-module of all linear maps from \(M\) to \(R\).

The even subalgebra of the Clifford algebra is defined as the submodule of the Clifford algebra \[ \mathcal {C}\kern -2pt\ell ^{+}(Q) \equiv \left \{ x_1 \cdots x_k \in \mathcal {C}\kern -2pt\ell \mid x \in V, k \text { is even} \right \} \] which also forms a subalgebra. Its elements are called even elements, as they can be expressed as the geometric product of an even number of 1-vectors.

Let \((\alpha , +_\alpha , *_\alpha )\) and \((\beta , +_\beta , *_\beta )\) be rings. A ring homomorphism from \(\alpha \) to \(\beta \) is a map \(\mathit {1} : \alpha \to _{+*} \beta \) such that

1. \(\mathit {1}(x +_{\alpha } y) = \mathit {1}(x) +_{\beta } \mathit {1}(y)\) for each \(x,y \in \alpha \).
2. \(\mathit {1}(x *_{\alpha } y) = \mathit {1}(x) *_{\beta } \mathit {1}(y)\) for each \(x,y \in \alpha \).
3. \(\mathit {1}(1_{\alpha }) = 1_{\beta }\).