Parity Multiplication is Associative/Proof 1

Theorem

Let $R := \struct {\set {\text{even}, \text{odd} }, +, \times}$ be the parity ring.

The operation $\times$ is associative:

$\forall a, b, c \in R: \paren {a \times b} \times c = a \times \paren {b \times c}$

Proof

From Isomorphism between Ring of Integers Modulo 2 and Parity Ring:

$\struct {\set {\text{even}, \text{odd} }, +, \times}$ is isomorphic with $\struct {\Z_2, +_2, \times_2}$

the ring of integers modulo $2$.

The result follows from:

Modulo Multiplication is Associative

and:

Isomorphism Preserves Associativity.

$\blacksquare$