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A [[Fractional ideal|fractional ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337301.png" /> of an integral commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337302.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337303.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337304.png" /> denotes the set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337305.png" /> of the [[field of fractions]] of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337306.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337307.png" />). A divisorial ideal is sometimes called a divisor of the ring. For any fractional ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337308.png" /> the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d0337309.png" /> is divisorial. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373010.png" /> of divisorial ideals of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373011.png" /> is a lattice-ordered commutative monoid (semi-group) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373012.png" /> is considered to be the product of two divisorial ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373014.png" />, while the integral divisorial ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373015.png" /> are considered as positive (or effective). The monoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373016.png" /> is a group if and only if the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373017.png" /> is completely integrally closed; in that case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373018.png" /> is the inverse of the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373019.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373020.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373022.png" /> are called quasi-equal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373023.png" />), which forms one of the principal subjects in algebra of these days — the study of factorization of ideals.
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Principal fractional ideals, as well as invertible fractional ideals, are divisorial and form subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373026.png" />, respectively. The quotient groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373028.png" /> are known, respectively, as the [[Divisor class group|divisor class group]] and the [[Picard group|Picard group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033730/d03373029.png" />.
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A [[Fractional ideal|fractional ideal]]  $  \mathfrak a $
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of an integral commutative ring  $  A $
 +
such that  $  \mathfrak a = A : ( A : \mathfrak a ) $(
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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 $
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is considered to be the product of two divisorial ideals  $  \mathfrak a $
 +
and  $  \mathfrak b $,
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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====
 
====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>
 
<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>

Revision as of 19:36, 5 June 2020


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)
How to Cite This Entry:
Divisorial ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divisorial_ideal&oldid=35052
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article