Isomorphism between Ring of Integers Modulo 2 and Parity Ring

Theorem
The ring of integers modulo $2$ and the parity ring are isomorphic.

Proof
To simplify the notation, let the elements of $\Z_2$ be identified as $0$ for $\eqclass 0 2$ and $1$ for $\eqclass 1 2$.

Let $f$ be the mapping from the parity ring $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ and the ring of integers modulo $2$ $\struct {\Z_2, +_2, \times_2}$:


 * $f: \struct {\set {\text{even}, \text{odd} }, +, \times} \to \struct {\Z_2, +_2, \times_2}$:
 * $\forall x \in R: \map f x = \begin{cases}

0 & : x = \text{even} \\ 1 & : x = \text{odd} \end{cases}$

The bijective nature of $f$ is apparent:
 * $f^{-1}: \struct {\Z_2, +_2, \times_2} \to \struct {\set {\text{even}, \text{odd} }, +, \times}$:
 * $\forall x \in \Z_2: \map {f^{-1} } x = \begin{cases}

\text{even} & : x = 0 \\ \text{odd} & : x = 1 \end{cases}$

Thus the following equations can be checked:

and:

These results can be determined from their Cayley tables: