Ring of rational integers

This article defines a particular commutative unital ring.
See all particular commutative unital rings

Definition

The ring $\mathbb {Z}$, called the ring of rational integers or sometimes simply the ring of integers, is the ring whose elements are the rational integers, with the usual addition and multiplication. Explicitly, the underlying set is $\{\dots ,-2,-1,0,1,2,\dots \}$ and the addition and multiplication are the usual ones.

The adjective rational is used in the name in circumstances where there may be potential confusion with the ring of integers in a number field.

This ring is the initial object in the category of commutative unital rings.

Ring properties

Property	Meaning	Satisfied?	Explanation
integral domain	zero is not a product of nonzero elements	Yes
Euclidean domain	admits a Euclidean norm	Yes	See ring of rational integers is Euclidean with norm equal to absolute value (the standard choice of norm is the absolute value); see also ring of rational integers is Euclidean with norm equal to binary logarithm of absolute value
principal ideal domain (PID)	every ideal is a principal ideal	Yes	Follows from being Euclidean and Euclidean implies PID
unique factorization domain	every element has a unique factorization into irreducibles (same as primes) up to units	Yes	Follows from being a PID and PID implies UFD
Noetherian domain	integral domain	Yes	Follows from being a PID
Bezout domain		Yes	Follows from being a PID
Dedekind domain		Yes	Follows from being a PID
interpolation domain		Yes	For any $n$, there exists a tuple of $n+1$ elements such that evaluation at these defines a bijection between the polynomials of degree at most $n$ in the ring of integer-valued polynomials and $R^{n+1}$.