Z2nZ

Z2nZ, typically written as Z/(2n)Z or Z_{2n}, denotes the ring of integers modulo 2n. It is the quotient of the ring of integers Z by the ideal 2nZ, and its elements can be represented by the residues 0, 1, ..., 2n−1. The arithmetic is performed with addition and multiplication performed modulo 2n.

As a mathematical object, Z/(2n)Z carries both a ring and an additive group structure. The additive group is cyclic of order 2n, generated by 1. The ring has a multiplicative identity, but it is not usually a field unless 2n is prime (which can occur only for n = 1, giving Z/2Z). In general, Z/(2n)Z contains zero divisors whenever 2n is not prime.

The units of Z/(2n)Z are precisely the elements coprime to 2n; their number is Euler’s totient φ(2n). The presence of zero divisors and units makes the ring a common setting for studies in modular arithmetic and number theory.

A key structural tool is the Chinese Remainder Theorem. If 2n factors as a product of coprime integers, say 2n = a b with gcd(a, b) = 1, then Z/(2n)Z ≅ Z/aZ × Z/bZ. More generally, Z/(2n)Z is isomorphic to the product of Z/p^kZ over the prime power factors p^k of 2n.

Ideals in Z/(2n)Z correspond to divisors of 2n: for each d dividing 2n, the ideal dZ/(2n)Z consists of all multiples of d modulo 2n.

Z2nZ is widely used in problems involving modular congruences, residue classes, and cryptographic constructions that rely on modular arithmetic.