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
The units of Z/(2n)Z are precisely the elements coprime to 2n; their number is Euler’s totient φ(2n).
A key structural tool is the Chinese Remainder Theorem. If 2n factors as a product of coprime
Ideals in Z/(2n)Z correspond to divisors of 2n: for each d dividing 2n, the ideal dZ/(2n)Z consists
Z2nZ is widely used in problems involving modular congruences, residue classes, and cryptographic constructions that rely