Divisorial ideal

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

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 $ D ( A) $. 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 $ \mathfrak a $ and $ \mathfrak b $ are called quasi-equal if $ \widetilde{\mathfrak a} = \widetilde{\mathfrak b} $), 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 $ F ( A) $ and $ J ( A) $ in $ D ( A) $, respectively. The quotient groups $ D ( A) / F ( A) = C ( A) $ and $ J ( A) / F ( A) = \mathop{\rm Pic} ( A) $ are known, respectively, as the divisor class group and the Picard group of $ A $.

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)