Ideal theory

<h2 id="ideals-in-a-finitely-generated-algebra-over-a-field">Ideals in a finitely generated algebra over a field</h2>

See also: <a href="/facts/Finitely_generated_algebra/57a93D5A">finitely generated algebra</a>
Ideals in a finitely generated algebra over a field (that is, a quotient of a <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> over a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, if 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a finitely generated algebra over a field, then the <a href="/facts/Radical_of_an_ideal/GYm3Pm9C">radical of an ideal</a> in 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is the intersection of all maximal ideals containing the ideal (because 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a <a href="/facts/Jacobson_ring/v6tX0FQP">Jacobson ring</a>). This may be thought of as an extension of <a href="/facts/Hilbert%2527s_Nullstellensatz/TlgoRHuM">Hilbert's Nullstellensatz</a>, which concerns the case when 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a polynomial ring.

<h2 id="topology-determined-by-an-ideal">Topology determined by an ideal</h2>
Main article: <a href="/facts/I-adic_topology/ALsc1Tc6">I-adic topology</a>
If I is an ideal in a ring A, then it determines the topology on A where a subset U of A is open if, for each x in U,

x
        +
        
          I
          
            n
          
        
        ⊂
        U
        .
      
    
    {\displaystyle x+I^{n}\subset U.}

for some integer 
 
 
 
 n
 >
 0
 
 
 {\displaystyle n>0}
 
. This topology is called the I-adic topology. It is also called an a-adic topology if 
 
 
 
 I
 =
 a
 A
 
 
 {\displaystyle I=aA}
 
 is generated by an element 
 
 
 
 a
 
 
 {\displaystyle a}
 
.
For example, take 
 
 
 
 A
 =
 
 Z
 
 
 
 {\displaystyle A=\mathbb {Z} }
 
, the ring of integers and 
 
 
 
 I
 =
 p
 A
 
 
 {\displaystyle I=pA}
 
 an ideal generated by a prime number p. For each integer 
 
 
 
 x
 
 
 {\displaystyle x}
 
, define 
 
 
 
 
 |
 
 x
 
 
 |
 
 
 p
 
 
 =
 
 p
 
 −
 n
 
 
 
 
 {\displaystyle |x|_{p}=p^{-n}}
 
 when 
 
 
 
 x
 =
 
 p
 
 n
 
 
 y
 
 
 {\displaystyle x=p^{n}y}
 
, 
 
 
 
 y
 
 
 {\displaystyle y}
 
 <a href="/facts/Coprime_integers/bnCxCXxs">prime to</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
. Then, clearly,

x
        +
        
          p
          
            n
          
        
        A
        =
        B
        (
        x
        ,
        
          p
          
            −
            (
            n
            −
            1
            )
          
        
        )
      
    
    {\displaystyle x+p^{n}A=B(x,p^{-(n-1)})}

where 
 
 
 
 B
 (
 x
 ,
 r
 )
 =
 {
 z
 ∈
 
 Z
 
 ∣
 
 |
 
 z
 −
 x
 
 
 |
 
 
 p
 
 
 <
 r
 }
 
 
 {\displaystyle B(x,r)=\{z\in \mathbb {Z} \mid |z-x|_{p}<r\}}
 
 denotes an open ball of radius 
 
 
 
 r
 
 
 {\displaystyle r}
 
 with center 
 
 
 
 x
 
 
 {\displaystyle x}
 
. Hence, the 
 
 
 
 p
 
 
 {\displaystyle p}
 
-adic topology on 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 is the same as the <a href="/facts/Metric_space/gnBBRXtc">metric space</a> topology given by 
 
 
 
 d
 (
 x
 ,
 y
 )
 =
 
 |
 
 x
 −
 y
 
 
 |
 
 
 p
 
 
 
 
 {\displaystyle d(x,y)=|x-y|_{p}}
 
. As a metric space, 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
 can be <a href="/facts/Completion_of_a_metric_space/lsckmU8G">completed</a>. The resulting complete metric space has a structure of a ring that extended the ring structure of 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
; this ring is denoted as 
 
 
 
 
 
 Z
 
 
 p
 
 
 
 
 {\displaystyle \mathbb {Z} _{p}}
 
 and is called the <a href="/facts/Ring_of_p-adic_integers/SDgDbkLF">ring of p-adic integers</a>.

<h2 id="ideal-class-group">Ideal class group</h2>
In a <a href="/facts/Dedekind_domain/tAsvAcaW">Dedekind domain</a> A (e.g., a ring of integers in a number field or the coordinate ring of a smooth affine curve) with the field of fractions 
 
 
 
 K
 
 
 {\displaystyle K}
 
, an ideal 
 
 
 
 I
 
 
 {\displaystyle I}
 
 is invertible in the sense: there exists a <a href="/facts/Fractional_ideal/I5rvJ6nt">fractional ideal</a> 
 
 
 
 
 I
 
 −
 1
 
 
 
 
 {\displaystyle I^{-1}}
 
 (that is, an A-submodule of 
 
 
 
 K
 
 
 {\displaystyle K}
 
) such that 
 
 
 
 I
 
 
 I
 
 −
 1
 
 
 =
 A
 
 
 {\displaystyle I\,I^{-1}=A}
 
, where the product on the left is a product of submodules of K. In other words, fractional ideals form a group under a product. The quotient of the group of fractional ideals by the subgroup of principal ideals is then the <a href="/facts/Ideal_class_group/LLn5Pwj1">ideal class group</a> of A.
In a general ring, an ideal may not be invertible (in fact, already the definition of a fractional ideal is not clear). However, over a Noetherian <a href="/facts/Integral_domain/cylt6zxW">integral domain</a>, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki's <a href="/facts/Alg%25C3%25A8bre_commutative/8TQroWxE">Algèbre commutative</a> gives such a theory.
The ideal class group of A, when it can be defined, is closely related to the <a href="/facts/Picard_group/lWH0CoWQ">Picard group</a> of the <a href="/facts/Prime_spectrum/2k2nDE5w">spectrum</a> of A (often the two are the same; e.g., for Dedekind domains).
In <a href="/facts/Algebraic_number/MkH1PzTK">algebraic number</a> theory, especially in <a href="/facts/Class_field_theory/4cvKbn6a">class field theory</a>, it is more convenient to use a generalization of an ideal class group called an <a href="/facts/Idele_class_group/lh1ccLJB">idele class group</a>.

<h2 id="closure-operations">Closure operations</h2>

There are several operations on ideals that play roles of closures. The most basic one is the <a href="/facts/Radical_of_an_ideal/GYm3Pm9C">radical of an ideal</a>. Another is the <a href="/facts/Integral_closure_of_an_ideal/MIGJSWHY">integral closure of an ideal</a>. Given an irredundant primary decomposition 
 
 
 
 I
 =
 ∩
 
 Q
 
 i
 
 
 
 
 {\displaystyle I=\cap Q_{i}}
 
, the intersection of 
 
 
 
 
 Q
 
 i
 
 
 
 
 {\displaystyle Q_{i}}
 
's whose radicals are minimal (don’t contain any of the radicals of other 
 
 
 
 
 Q
 
 j
 
 
 
 
 {\displaystyle Q_{j}}
 
's) is uniquely determined by 
 
 
 
 I
 
 
 {\displaystyle I}
 
; this intersection is then called the unmixed part of 
 
 
 
 I
 
 
 {\displaystyle I}
 
. It is also a closure operation.
Given ideals 
 
 
 
 I
 ,
 J
 
 
 {\displaystyle I,J}
 
 in a ring 
 
 
 
 A
 
 
 {\displaystyle A}
 
, the ideal

(
        I
        :
        
          J
          
            ∞
          
        
        )
        =
        {
        f
        ∈
        A
        ∣
        f
        
          J
          
            n
          
        
        ⊂
        I
        ,
        n
        ≫
        0
        }
        =
        
          ⋃
          
            n
            >
            0
          
        
        
          Ann
          
            A
          
        
        ⁡
        (
        (
        
          J
          
            n
          
        
        +
        I
        )
        
          /
        
        I
        )
      
    
    {\displaystyle (I:J^{\infty })=\{f\in A\mid fJ^{n}\subset I,n\gg 0\}=\bigcup _{n>0}\operatorname {Ann} _{A}((J^{n}+I)/I)}

is called the saturation of 
 
 
 
 I
 
 
 {\displaystyle I}
 
 with respect to 
 
 
 
 J
 
 
 {\displaystyle J}
 
 and is a closure operation (this notion is closely related to the study of local cohomology).
See also <a href="/facts/Tight_closure/w166f3bQ">tight closure</a>.

<h2 id="reduction-theory">Reduction theory</h2>
Main article: <a href="/facts/Ideal_reduction/rrPU6LNs">Ideal reduction</a>
<h2 id="local-cohomology-in-ideal-theory">Local cohomology in ideal theory</h2>

Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory.
Let 
 
 
 
 M
 
 
 {\displaystyle M}
 
 be a module over a ring 
 
 
 
 R
 
 
 {\displaystyle R}
 
 and 
 
 
 
 I
 
 
 {\displaystyle I}
 
 an ideal. Then 
 
 
 
 M
 
 
 {\displaystyle M}
 
 determines the sheaf 
 
 
 
 
 
 
 M
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {M}}}
 
 on 
 
 
 
 Y
 =
 Spec
 ⁡
 (
 R
 )
 −
 V
 (
 I
 )
 
 
 {\displaystyle Y=\operatorname {Spec} (R)-V(I)}
 
 (the restriction to Y of the sheaf associated to M). Unwinding the definition, one sees:

Γ
 
 I
 
 
 (
 M
 )
 :=
 Γ
 (
 Y
 ,
 
 
 
 M
 ~
 
 
 
 )
 =
 
 
 lim
 →
 
 
 ⁡
 Hom
 ⁡
 (
 
 I
 
 n
 
 
 ,
 M
 )
 
 
 {\displaystyle \Gamma _{I}(M):=\Gamma (Y,{\widetilde {M}})=\varinjlim \operatorname {Hom} (I^{n},M)}
 
.
Here, 
 
 
 
 
 Γ
 
 I
 
 
 (
 M
 )
 
 
 {\displaystyle \Gamma _{I}(M)}
 
 is called the ideal transform of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 with respect to 
 
 
 
 I
 
 
 {\displaystyle I}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/System_of_parameters/ALsc1Tc6">System of parameters</a></li></ul>

<ul><li><a href="/facts/Michael_Atiyah/RQVJ9ATE">Atiyah, Michael Francis</a>; <a href="/facts/Ian_G._Macdonald/PoaeNbeg">Macdonald, I.G.</a> (1969), Introduction to Commutative Algebra, Westview Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-201-40751-8</li>
<li><a href="/facts/David_Eisenbud/hw2yEnnx">Eisenbud, David</a>, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-94268-8.</li>
<li>Huneke, Craig; <a href="/facts/Irena_Swanson/53TBTHWn">Swanson, Irena</a> (2006), <a href="https://web.archive.org/web/20191115053353/http://people.reed.edu/~iswanson/book/index.html">Integral closure of ideals, rings, and modules</a>, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-68860-4, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2266432">2266432</a>, archived from <a href="http://people.reed.edu/~iswanson/book/index.html">the original</a> on 2019-11-15, retrieved 2019-11-15</li></ul>

Ideal theory open-in-new

Ideal theory