linguistic:translation
Definition

An even number is an integer of the form x = 2 k {\displaystyle x=2k} where k is an integer; an odd number is an integer of the form x = 2 k + 1. {\displaystyle x=2k+1.}

An equivalent definition is that an even number is divisible by 2: 2 | x {\displaystyle 2\ |\ x} and an odd number is not: 2 ⧸ | x {\displaystyle 2\not |\ x}

The sets of even and odd numbers can be defined as following: { 2 k: k ∈ Z } {\displaystyle \{2k:k\in \mathbb {Z} \}} { 2 k + 1: k ∈ Z } {\displaystyle \{2k+1:k\in \mathbb {Z} \}}

The set of even numbers is a prime ideal of Z {\displaystyle \mathbb {Z} } and the quotient ring Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } is the field with two elements. Parity can then be defined as the unique ring homomorphism from Z {\displaystyle \mathbb {Z} } to Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } where odd numbers are 1 and even numbers are 0. The consequences of this homomorphism are covered below.