nLab integers modulo n

Definition

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

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

Properties

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

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