# Ring of rational integers

From Commalg

This article defines a particular commutative unital ring.

See all particular commutative unital rings

## Definition

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.

## 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 , 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 . |