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}$

Also see

• Parity Ring is Ring
• Parity Ring is Smallest Field
• Isomorphism between Ring of Integers Modulo 2 and Parity Ring

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.2$