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A [[Fractional ideal|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 [[#References|[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|divisor class group]] and the [[Picard group|Picard group]] of  $  A $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>