Parity Multiplication is Commutative/Proof 1

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Theorem

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


The operation $\times$ is commutative:

$\forall a, b \in R: a \times b = b \times a$


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 Commutative

and:

Isomorphism Preserves Associativity.

$\blacksquare$