pZ

pZ denotes the set of all integer multiples of p: pZ = { pk : k ∈ Z }. It is a subgroup of the additive group of integers and the principal ideal generated by p in the ring Z.

If p = 0, pZ = {0}. For p ≠ 0, pZ has smallest positive element |p|, and the index [Z : pZ] equals p. The quotient Z/pZ is a cyclic group of order p, and Z/(pZ) ≅ Z/pZ. The natural projection π: Z → Z/pZ has kernel pZ.

The map n ↦ pn provides an isomorphism from Z to pZ as additive groups; thus pZ ≅ Z as abelian groups. Note that pZ is not a unital subring of Z; it is an ideal of Z.

For two nonzero integers p and q, the ideals they generate are (p) and (q). Then (p) ∩ (q) = lcm(p,q)Z and (p) + (q) = gcd(p,q)Z.

Examples: 3Z = { ..., -6, -3, 0, 3, 6, ... }; 5Z = { ..., -10, -5, 0, 5, 10, ... }.

Relation to primes: If p is prime, the quotient Z/pZ is a field, and the ideal pZ is a maximal ideal of Z. In summary, pZ is the principal, non-unital ideal of Z consisting of all multiples of p, with properties closely tied to modular arithmetic and the structure of the integers as a principal ideal domain.