Ring of rational integers
This article defines a particular commutative unital ring.
See all particular commutative unital rings
The ring , 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 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.
|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 , there exists a tuple of elements such that evaluation at these defines a bijection between the polynomials of degree at most in the ring of integer-valued polynomials and .|