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) |
Divisorial ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divisorial_ideal&oldid=13538