# Definition:Parity Ring

## Definition

The parity ring is the ring of two elements which defines the nature of the parity of integers under addition and multiplication:

$\struct {\set {\text{even}, \text{odd} }, +, \times}$

## Cayley Tables

The parity ring can be described completely by showing its Cayley tables:

$\begin{array}{r|rr} + & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} \\ \end{array} \qquad \begin{array}{r|rr} \times & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{even} \\ \text{odd} & \text{even} & \text{odd} \\ \end{array}$