# 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 $\widetilde{\mathfrak a \cdot \mathfrak b }$ 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 , 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 $A$.