qZ

qZ denotes the set of all integer multiples of q, where q is an integer. Formally, qZ = { qk : k ∈ Z }. For q ≠ 0, this is an infinite cyclic subgroup of the additive group of integers and, in ring-theory terms, a principal ideal of the ring Z generated by q. If q = 0, then qZ = {0}.

In more detail, qZ is closed under addition and subtraction and is the kernel of the natural reduction map from Z to Z/qZ. The quotient Z/qZ is a finite cyclic group (and ring) of order |q| when q ≠ 0. Consequently, the index of qZ in Z is |q|, meaning there are exactly |q| distinct cosets of qZ in Z. For q = ±1, qZ equals Z itself, and Z/qZ is the trivial group.

Examples help illustrate the concept. If q = 3, then 3Z = {..., -6, -3, 0, 3, 6, ...}, the set of multiples of 3. The quotient Z/3Z consists of the residue classes [0], [1], and [2], forming a cyclic group of order 3. If q = 0, 0Z = {0} and Z/0Z is isomorphic to Z, an infinite group.

Related ideas include principal ideals in Z, cyclic groups, and modular arithmetic. The notation qZ is widely used to discuss divisibility, residue classes, and the structure of subgroups and ideals within the integers.