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An associative-commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305501.png" /> with a unit having no divisors of zero (i.e. a commutative integral domain) in which each proper ideal can be represented as a product of prime ideals (an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305502.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305503.png" /> is said to be prime if the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305504.png" /> does not contain divisors of zero). This name was given to such rings in honour of R. Dedekind, who was one of the first to study such rings in the 1870s.
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Any principal ideal domain is a Dedekind ring. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305505.png" /> is a Dedekind ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305506.png" /> is a finite algebraic extension of its quotient field, the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305507.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305508.png" /> (i.e. the totality of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d0305509.png" /> which are roots of equations of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055011.png" />) will again be a Dedekind ring. In particular, one has that the rings of algebraic integers and maximal orders in algebraic number fields are Dedekind, i.e. the integral closures of rings of integers in finite algebraic extensions of the field of rational numbers.
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In a Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055012.png" /> each proper ideal can be uniquely represented as a product of prime ideals. This theorem arose in the problem of decomposition of elements into prime factors in maximal orders in algebraic number fields. Such a decomposition is, as a rule, not unique.
+
An associative-commutative ring  $  R $
 +
with a unit having no divisors of zero (i.e. a commutative integral domain) in which each proper ideal can be represented as a product of prime ideals (an ideal  $  P $
 +
of  $  R $
 +
is said to be prime if the quotient ring  $  R / P $
 +
does not contain divisors of zero). This name was given to such rings in honour of R. Dedekind, who was one of the first to study such rings in the 1870s.
  
A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055013.png" /> is a Dedekind ring if and only if the semi-group of fractional ideals (cf. [[Fractional ideal|Fractional ideal]]) of this ring is a group. Each fractional ideal of a Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055014.png" /> can be uniquely represented as the product of (positive or negative) powers of prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055015.png" />. A Dedekind ring can be characterized as follows: A commutative integral domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055016.png" /> is a Dedekind ring if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055017.png" /> is a [[Noetherian ring|Noetherian ring]], if each proper prime ideal of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055018.png" /> is maximal and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055019.png" /> is integrally closed, i.e. coincides with its integral closure in its field of fractions. In other words, a Dedekind ring is a Noetherian normal ring of Krull dimension one. The so-called Chinese remainder theorem is valid for a Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055020.png" />: For a given finite selection of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055021.png" /> and elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055024.png" />, the system of congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055026.png" />) has a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055027.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055028.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055029.png" />) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055030.png" />. A Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055031.png" /> may also be characterized as a [[Krull ring|Krull ring]] of dimension one. Each Dedekind ring is a regular commutative ring and all its localizations by maximal ideals form a [[Discretely-normed ring|discretely-normed ring]]. The semi-group of non-zero ideals of a Dedekind ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055032.png" /> is isomorphic to the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030550/d03055033.png" /> of divisors (cf. [[Divisor|Divisor]]) of this ring.
+
Any principal ideal domain is a Dedekind ring. If  $  R $
 +
is a Dedekind ring and  $  L $
 +
is a finite algebraic extension of its quotient field, the integral closure of  $  R $
 +
in  $  L $(
 +
i.e. the totality of elements of  $  L $
 +
which are roots of equations of the type  $  x  ^ {n} + a _ {1} x  ^ {n-} 1 + \dots + a _ {n} = 0 $,
 +
$  a _ {i} \in R $)
 +
will again be a Dedekind ring. In particular, one has that the rings of algebraic integers and maximal orders in algebraic number fields are Dedekind, i.e. the integral closures of rings of integers in finite algebraic extensions of the field of rational numbers.
 +
 
 +
In a Dedekind ring  $  R $
 +
each proper ideal can be uniquely represented as a product of prime ideals. This theorem arose in the problem of decomposition of elements into prime factors in maximal orders in algebraic number fields. Such a decomposition is, as a rule, not unique.
 +
 
 +
A ring  $  R $
 +
is a Dedekind ring if and only if the semi-group of fractional ideals (cf. [[Fractional ideal|Fractional ideal]]) of this ring is a group. Each fractional ideal of a Dedekind ring $  R $
 +
can be uniquely represented as the product of (positive or negative) powers of prime ideals of $  R $.  
 +
A Dedekind ring can be characterized as follows: A commutative integral domain $  R $
 +
is a Dedekind ring if and only if $  R $
 +
is a [[Noetherian ring|Noetherian ring]], if each proper prime ideal of the ring $  R $
 +
is maximal and if $  R $
 +
is integrally closed, i.e. coincides with its integral closure in its field of fractions. In other words, a Dedekind ring is a Noetherian normal ring of Krull dimension one. The so-called Chinese remainder theorem is valid for a Dedekind ring $  R $:  
 +
For a given finite selection of ideals $  I _ {i} $
 +
and elements $  x _ {i} $
 +
of $  R $,  
 +
$  i= 1 \dots n $,  
 +
the system of congruences $  x \equiv x _ {i} $(
 +
$  \mathop{\rm mod}  I _ {i} $)  
 +
has a solution $  x \in R $
 +
if and only if $  x _ {i} \equiv x _ {j} $(
 +
$  \mathop{\rm mod}  I _ {i} + I _ {j} $)  
 +
for $  i \neq j $.  
 +
A Dedekind ring $  R $
 +
may also be characterized as a [[Krull ring|Krull ring]] of dimension one. Each Dedekind ring is a regular commutative ring and all its localizations by maximal ideals form a [[Discretely-normed ring|discretely-normed ring]]. The semi-group of non-zero ideals of a Dedekind ring $  R $
 +
is isomorphic to the semi-group $  P $
 +
of divisors (cf. [[Divisor|Divisor]]) of this ring.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[4]</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">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For Krull dimension see (the editorial comments to) [[Dimension|Dimension]]. For normal ring see [[Normal ring|Normal ring]].
 
For Krull dimension see (the editorial comments to) [[Dimension|Dimension]]. For normal ring see [[Normal ring|Normal ring]].

Latest revision as of 17:32, 5 June 2020


An associative-commutative ring $ R $ with a unit having no divisors of zero (i.e. a commutative integral domain) in which each proper ideal can be represented as a product of prime ideals (an ideal $ P $ of $ R $ is said to be prime if the quotient ring $ R / P $ does not contain divisors of zero). This name was given to such rings in honour of R. Dedekind, who was one of the first to study such rings in the 1870s.

Any principal ideal domain is a Dedekind ring. If $ R $ is a Dedekind ring and $ L $ is a finite algebraic extension of its quotient field, the integral closure of $ R $ in $ L $( i.e. the totality of elements of $ L $ which are roots of equations of the type $ x ^ {n} + a _ {1} x ^ {n-} 1 + \dots + a _ {n} = 0 $, $ a _ {i} \in R $) will again be a Dedekind ring. In particular, one has that the rings of algebraic integers and maximal orders in algebraic number fields are Dedekind, i.e. the integral closures of rings of integers in finite algebraic extensions of the field of rational numbers.

In a Dedekind ring $ R $ each proper ideal can be uniquely represented as a product of prime ideals. This theorem arose in the problem of decomposition of elements into prime factors in maximal orders in algebraic number fields. Such a decomposition is, as a rule, not unique.

A ring $ R $ is a Dedekind ring if and only if the semi-group of fractional ideals (cf. Fractional ideal) of this ring is a group. Each fractional ideal of a Dedekind ring $ R $ can be uniquely represented as the product of (positive or negative) powers of prime ideals of $ R $. A Dedekind ring can be characterized as follows: A commutative integral domain $ R $ is a Dedekind ring if and only if $ R $ is a Noetherian ring, if each proper prime ideal of the ring $ R $ is maximal and if $ R $ is integrally closed, i.e. coincides with its integral closure in its field of fractions. In other words, a Dedekind ring is a Noetherian normal ring of Krull dimension one. The so-called Chinese remainder theorem is valid for a Dedekind ring $ R $: For a given finite selection of ideals $ I _ {i} $ and elements $ x _ {i} $ of $ R $, $ i= 1 \dots n $, the system of congruences $ x \equiv x _ {i} $( $ \mathop{\rm mod} I _ {i} $) has a solution $ x \in R $ if and only if $ x _ {i} \equiv x _ {j} $( $ \mathop{\rm mod} I _ {i} + I _ {j} $) for $ i \neq j $. A Dedekind ring $ R $ may also be characterized as a Krull ring of dimension one. Each Dedekind ring is a regular commutative ring and all its localizations by maximal ideals form a discretely-normed ring. The semi-group of non-zero ideals of a Dedekind ring $ R $ is isomorphic to the semi-group $ P $ of divisors (cf. Divisor) of this ring.

References

[1] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[2] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
[3] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)
[4] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)

Comments

For Krull dimension see (the editorial comments to) Dimension. For normal ring see Normal ring.

How to Cite This Entry:
Dedekind ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_ring&oldid=46598
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article