Zq

Zq, often written as Z/qZ or ℤ/qℤ, denotes the ring of integers modulo q. It consists of the q residue classes {0, 1, ..., q−1} with addition and multiplication performed modulo q. As an additive group, Zq is cyclic of order q, generated by 1. As a ring, it is commutative with a multiplicative identity, but its structure depends on q: if q is prime, Zq is a finite field; if q is composite, Zq contains zero divisors and is not a field.

When q is prime, the nonzero elements of Zq form a cyclic multiplicative group of order q−1. In general, the set of units in Zq—elements a with gcd(a, q) = 1—has φ(q) elements, where φ is Euler’s totient function. The ring Zq is a useful model for modular arithmetic, congruences, and number-theoretic algorithms.

Relation to other notation is context-dependent. Zq is often used interchangeably with Z/qZ or ℤ/qℤ, but in different areas, related notations appear: Fq or GF(q) denotes a finite field with q elements, which exists whenever q is a prime power; Zp, with p prime, is sometimes used to denote p-adic integers in a different mathematical setting. Therefore, while Zq commonly refers to the integers modulo q, one should consider context to distinguish between modular arithmetic, finite fields, and p-adic constructions.