2z

2Z denotes the set of all even integers. It is defined as {2k | k ∈ Z} and is a subset of the integers under addition.

In algebraic terms, 2Z is the principal ideal generated by 2 in the ring Z, and it forms an additive subgroup of Z. It is closed under addition and negation, and the product of any two elements in 2Z is also in 2Z. The quotient Z/2Z has two elements, corresponding to the even and odd classes, so the index of 2Z in Z is 2.

Cosets of 2Z partition Z into even and odd numbers. The cosets are 2Z itself and 1 + 2Z, where 1 + 2Z consists of all odd integers. This mirrors the fact that every integer is either even or odd.

As a ring-theoretic object, 2Z is a subring of Z (though it lacks a multiplicative identity, since 1 ∉ 2Z). It is also a prime ideal of Z, because the quotient Z/2Z is a field. More generally, 2Z is the case nZ for n = 2: the set of multiples of n, which in turn is a fundamental example of a principal ideal.

Equivalently, 2Z is isomorphic to Z as an additive group via the map k ↦ 2k, indicating that 2Z has the same infinite cyclic structure as Z. In number theory, 2Z is used to describe evenness, parity arguments, and modular arithmetic modulo 2.