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 $\left[\!\left[{0}\right]\!\right]_2$ and $1$ for $\left[\!\left[{1}\right]\!\right]_2$.

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


 * $f: \left({\left\{ {\text{even}, \text{odd} }\right\}, +, \times}\right) \to \left({\Z_2, +_2, \times_2}\right)$:
 * $\forall x \in R: f \left({x}\right) = \begin{cases}

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

The bijective nature of $f$ is apparent:
 * $f^{-1}: \left({\Z_2, +_2, \times_2}\right) \to \left({\left\{ {\text{even}, \text{odd} }\right\}, +, \times}\right)$:
 * $\forall x \in \Z_2: f^{-1} \left({x}\right) = \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: