integers modulo n in nLab

Contents

Definition

Given a natural number nn, the ring of integers modulo nn is the quotient ring ℤ/nℤ\mathbb{Z}/n\mathbb{Z}.

These are also the quotient rig ℕ/nℕ\mathbb{N}/n\mathbb{N}.

Properties

The integers modulo nn are precisely the finite cyclic rings, since the underlying set is the finite set of cardinality nn and the underlying abelian group is the cyclic group of order nn.

Given any positive integer nn, ℤ/nℤ\mathbb{Z}/n\mathbb{Z} is a prefield ring whose monoid of cancellative elements consists of all integers mm modulo nn which are coprime with nn. For nn a prime number this is a prime field, and for nn a prime power this is a prime power local ring.

See also

• prime field

• prime power local ring

• multiplicative group of integers modulo n

• modular polynomial

References

e.g. example 5 in these notes: pdf

 External links

• Wikipedia, Modular arithmetic#Integers modulo n