# Divisorial ideal

A fractional ideal of an integral commutative ring such that (here denotes the set of elements of the field of fractions of the ring for which ). A divisorial ideal is sometimes called a divisor of the ring. For any fractional ideal the ideal is divisorial. The set of divisorial ideals of the ring is a lattice-ordered commutative monoid (semi-group) if is considered to be the product of two divisorial ideals and , while the integral divisorial ideals are considered as positive (or effective). The monoid is a group if and only if the ring is completely integrally closed; in that case, is the inverse of the divisor .

Divisorial ideals are usually considered in a Krull ring (e.g. in a Noetherian integrally closed ring); here, prime ideals of height 1 are divisorial and form a basis of the Abelian group of divisors . This result is in fact due to E. Artin and B.L. van der Waerden [1], and forms part of their theory of quasi-equality of ideals (two ideals and are called quasi-equal if ), which forms one of the principal subjects in algebra of these days — the study of factorization of ideals.

Principal fractional ideals, as well as invertible fractional ideals, are divisorial and form subgroups and in , respectively. The quotient groups and are known, respectively, as the divisor class group and the Picard group of .

#### References

[1] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) |

[2] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |

**How to Cite This Entry:**

Divisorial ideal.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Divisorial_ideal&oldid=35052