3Z

3Z denotes the set of all multiples of three within the integers. If Z represents the set of all integers, then 3Z = { ..., -6, -3, 0, 3, 6, ... }. In algebra, 3Z is an additive subgroup of Z and an ideal of the ring Z, generated by 3. It plays a role in modular arithmetic and in the study of divisibility, serving as a basic example of a principal ideal.

The set 3Z yields a natural partition of the integers into three congruence classes modulo 3: 3Z, 3Z+1, and 3Z+2. The quotient ring Z/3Z has three elements and, because 3 is prime, forms a field isomorphic to the finite field with three elements (GF(3)). This construction underpins simple arithmetic modulo 3 and appears throughout number theory and algebra.

Outside pure mathematics, the string "3Z" can appear as a label, model number, or code in various domains. It may be used in product naming, catalog identifiers, or media references, among other contexts. Because 3Z is not tied to a single fixed entity, its meaning depends on the specific system or discipline in which it is encountered.

See also: integers, multiples of a number, modular arithmetic, quotient structures, and principal ideals.