Ideal theory: Difference between revisions
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{{short description|Theory of ideals in commutative rings in mathematics}} |
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{{about|the mathematical theory|the usage in political philosophy|Ideal theory (politics)}} |
{{about|the mathematical theory|the usage in political philosophy|Ideal theory (politics)}} |
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In [[mathematics]], '''ideal theory''' is the theory of [[ideal (ring theory)|ideal]]s in [[commutative ring]]s. While the notion of an ideal exists also for [[Noncommutative ring|non-commutative rings]], a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.) |
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Throughout the articles, rings refer to commutative rings. See also the article [[ideal (ring theory)]] for basic operations such as sum or products of ideals. |
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In [[mathematics]], '''ideal theory''' is the theory of [[ideal (ring theory)|ideal]]s in [[commutative ring]]s; and is the precursor name for the contemporary subject of [[commutative algebra]]. The name grew out of the central considerations, such as the [[Lasker–Noether theorem]] in [[algebraic geometry]], and the [[ideal class group]] in [[algebraic number theory]], of the commutative algebra of the first quarter of the twentieth century. It was used in the influential [[Bartel Leendert van der Waerden|van der Waerden]] text on [[abstract algebra]] from around 1930. |
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== Ideals in a finitely generated algebra over a field == |
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The ideal theory in question had been based on [[elimination theory]], but in line with [[David Hilbert]]'s taste moved away from [[algorithm]]ic methods. [[Gröbner basis]] theory has now reversed the trend, for [[computer algebra]]. |
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{{expand section|date=May 2022}} |
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{{see also|finitely generated algebra}} |
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Ideals in a finitely generated algebra over a field (that is, a quotient of a polynomial ring over a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, if <math>A</math> is a finitely generated algebra over a field, then the [[radical of an ideal]] in <math>A</math> is the intersection of all maximal ideals containing the ideal (because <math>A</math> is a [[Jacobson ring]]). This may be thought of as an extension of [[Hilbert's Nullstellensatz]], which concerns the case when <math>A</math> is a polynomial ring.<!--On the other hand, the [[Noether normalization lemma]]--> |
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The importance of the idea of a [[module (mathematics)|module]], more general than an ''ideal'', probably led to the perception that ''ideal theory'' was too narrow a description. [[Valuation theory]], too, was an important technical extension, and was used by [[Helmut Hasse]] and [[Oscar Zariski]]. [[Nicolas Bourbaki|Bourbaki]] used ''commutative algebra''; sometimes ''local algebra'' is applied to the theory of [[local ring]]s. [[Douglas Northcott]]'s 1953 [[Cambridge Tract]] ''Ideal Theory'' (reissued 2004 under the same title) was one of the final appearances of the name. |
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== Topology determined by an ideal == |
== Topology determined by an ideal == |
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{{main|I-adic topology}} |
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Let ''R'' be a ring and ''M'' an ''R''-module. Then each ideal <math>\mathfrak{a}</math> of ''R'' determines a topology on ''M'' called the <math>\mathfrak{a}</math>-adic topology such that a subset ''U'' of ''M'' is [[open subset|open]] if and only if for each ''x'' in ''U'' there exists a positive integer ''n'' such that |
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With respect to this <math>\mathfrak{a}</math>-adic topology, <math>\{x + \mathfrak{a}^n M\}_{n}</math> is a basis of neighbourhoods of <math>x</math> and makes the module operations continuous; in particular, <math>M</math> is a possibly non-Hausdorff [[topological group]]. Also, ''M'' is a [[Hausdorff topological space]] if and only if <math display="inline">\bigcap_{n > 0} \mathfrak{a}^nM = 0.</math> Moreover, when <math>M</math> is Hausdorff, the topology is the same as the [[metric space]] topology given by defining the distance function: <math>d(x, y) = 2^{-n}</math> for <math>x \ne y</math>, where <math>n</math> is an integer such that <math>x - y \in \mathfrak{a}^n M - \mathfrak{a}^{n+1} M</math>. |
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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'', |
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Given a submodule ''N'' of ''M'', the <math>\mathfrak{a}</math>-closure of ''N'' in ''M'' is equal to <math display="inline">\bigcap_{n > 0} (N + \mathfrak{a}^n M)</math>, as shown easily. |
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for some integer <math>n > 0</math>. This topology is called the ''I''-adic topology. It is also called an ''a''-adic topology if <math>I = aA</math> is generated by an element <math>a</math>. |
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For example, take <math>A = \mathbb{Z}</math>, the ring of integers and <math>I = pA</math> an ideal generated by a prime number ''p''. For each integer <math>x</math>, define <math>|x|_p = p^{-n}</math> when <math>x = p^n y</math>, <math>y</math> [[Coprime integers|prime to]] <math>p</math>. Then, clearly, |
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:<math>x + p^n A = B(x, p^{-(n-1)})</math> |
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where <math>B(x, r) = \{ z \in \mathbb{Z} \mid |z - x|_p < r \}</math> denotes an open ball of radius <math>r</math> with center <math>x</math>. Hence, the <math>p</math>-adic topology on <math>\mathbb{Z}</math> is the same as the [[metric space]] topology given by <math>d(x, y) = |x - y|_p</math>. As a metric space, <math>\mathbb{Z}</math> can be [[completion of a metric space|completed]]. The resulting complete metric space has a structure of a ring that extended the ring structure of <math>\mathbb{Z}</math>; this ring is denoted as <math>\mathbb{Z}_p</math> and is called the [[ring of p-adic integers|ring of ''p''-adic integers]]. |
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== Ideal class group == |
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When <math>M</math> is Hausdorff, <math>M</math> can be [[completion of a metric space|completed]] as a metric space; the resulting space is denoted by <math>\widehat{M}</math> and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): |
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In a [[Dedekind domain]] ''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 <math>K</math>, an ideal <math>I</math> is invertible in the sense: there exists a [[fractional ideal]] <math>I^{-1}</math> (that is, an ''A''-submodule of <math>K</math>) such that <math>I \, I^{-1} = A</math>, 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 [[ideal class group]] of ''A''. |
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:<math>\widehat{M} = \varprojlim M/\mathfrak{a}^n M</math> |
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where the right-hand side is the [[completion (algebra)|completion]] of the module <math>M</math> with respect to <math>\mathfrak{a}</math>. |
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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 integral domain, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki's ''[[Algèbre commutative]]'' gives such a theory.<!-- give the details of the theory --> |
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'''Example''': Let <math>R = k[x_1, \dots, x_n]</math> be a polynomial ring over a field and <math>\mathfrak{a} = (x_1, \dots, x_n)</math> the maximal ideal. Then <math>\widehat{R} = k [\![x_1, \dots, x_n]\!]</math> is a [[formal power series ring]].<!-- More generally, ... --> |
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The ideal class group of ''A'', when it can be defined, is closely related to the [[Picard group]] of the [[prime spectrum|spectrum]] of ''A'' (often the two are the same; e.g., for Dedekind domains). |
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''R'' is called a [[Zariski ring]] with respect to <math>\mathfrak{a}</math> if every ideal in ''R'' is <math>\mathfrak{a}</math>-closed. There is a characterization: |
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:''R'' is a Zariski ring with respect to <math>\mathfrak{a}</math> if and only if <math>\mathfrak{a}</math> is contained in the [[Jacobson radical]] of ''R''. |
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In particular a Noetherian local ring is a Zariski ring with respect to the maximal ideal. |
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In algebraic number theory, especially in [[class field theory]], it is more convenient to use a generalization of an ideal class group called an [[idele class group]]. |
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A '''system of parameters''' for a [[local ring|local]] [[Noetherian ring]] of [[Krull dimension]] ''d'' with [[maximal ideal]] ''m'' is a set of elements ''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub> that satisfies any of the following equivalent conditions: |
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# ''m'' is a [[Minimal prime ideal|minimal prime]] over (''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>). |
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# The [[radical of an ideal|radical]] of (''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>) is ''m''. |
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# Some power of ''m'' is contained in (''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>). |
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# (''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>) is [[primary ideal|''m''-primary]]. |
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Every local Noetherian ring admits a system of parameters. |
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== Closure operations == |
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It is not possible for fewer than ''d'' elements to generate an ideal whose radical is ''m'' because then the dimension of ''R'' would be less than ''d''. |
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{{expand section|date=May 2022}} |
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There are several operations on ideals that play roles of closures. The most basic one is the [[radical of an ideal]]. Another is the [[integral closure of an ideal]]. Given an irredundant primary decomposition <math>I = \cap Q_i</math>, the intersection of <math>Q_i</math>'s whose radicals are minimal (don’t contain any of the radicals of other <math>Q_j</math>'s) is uniquely determined by <math>I</math>; this intersection is then called the unmixed part of <math>I</math>. It is also a closure operation. |
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If ''M'' is a ''k''-dimensional module over a local ring, then ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub> is a '''system of parameters''' for ''M'' if the [[Length_of_a_module|length]] of {{nowrap|''M'' / (''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>)''M''}} is finite. |
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Given ideals <math>I, J</math> in a ring <math>A</math>, the ideal |
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:<math>(I : J^{\infty}) = \{ f \in A \mid fJ^n \subset I, n \gg 0 \} = \bigcup_{n > 0} \operatorname{Ann}_A((J^n + I)/I)</math> |
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The reduction theory goes back to the influential 1954 paper by Northcott and Rees, the paper that introduced the basic notions. In algebraic geometry, the theory is among the essential tools to extract detailed information about the behaviors of [[blowing up|blow-up]]s. |
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is called the saturation of <math>I</math> with respect to <math>J</math> and is a closure operation (this notion is closely related to the study of local cohomology). |
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See also [[tight closure]]. |
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Given ideals ''J'' ⊂ ''I'' in a ring ''R'', the ideal ''J'' is said to be a ''reduction'' of ''I'' if there is some integer ''m'' > 0 such that <math>JI^m = I^{m+1}</math>.<ref>{{harvnb|Huneke|Swanson|2006|loc=Definition 1.2.1}}</ref> For such ideals, immediately from the definition, the following hold: |
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*For any ''k'', <math>J^k I^m = J^{k-1}I^{m+1} = \cdots = I^{m+k}</math>. |
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*''J'' and ''I'' have the same radical and the same set of minimal prime ideals over them<ref>{{harvnb|Huneke|Swanson|2006|loc=Lemma 8.1.10}}</ref> (the converse is false). |
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If ''R'' is a Noetherian ring, then ''J'' is a reduction of ''I'' if and only if the [[Rees algebra]] ''R''[''It''] is [[finitely generated module|finite]] over ''R''[''Jt''].<ref>{{harvnb|Huneke|Swanson|2006|loc=Theorem 8.2.1.}}</ref> (This is the reason for the relation to a blow up.) |
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{{main|Ideal reduction}} |
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A closely related notion is that of '''analytic spread'''. By definition, the '''fiber cone ring''' of a Noetherian local ring (''R'', <math>\mathfrak{m}</math>) along an ideal ''I'' is |
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:<math>\mathcal{F}_I(R) = R[It] \otimes_R \kappa(\mathfrak{m}) \simeq \bigoplus_{n=0}^{\infty} I^n/\mathfrak{m} I^n</math>. |
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The [[Krull dimension]] of <math>\mathcal{F}_I(R)</math> is called the ''analytic spread'' of ''I''. Given a reduction <math>J \subset I</math>, the minimum number of generators of ''J'' is at least the analytic spread of ''I''.<ref>{{harvnb|Huneke|Swanson|2006|loc=Corollary 8.2.5.}}</ref> Also, a partial converse holds for infinite fields: if <math>R/\mathfrak m</math> is infinite and if the integer <math>\ell</math> is the analytic spread of ''I'', then each reduction of ''I'' contains a reduction generated by <math>\ell</math> elements.<ref>{{harvnb|Huneke|Swanson|2006|loc=Proposition 8.3.7}}</ref> |
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== Local cohomology in ideal theory == |
== Local cohomology in ideal theory == |
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{{expand section}} |
{{expand section|date=December 2019}} |
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Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory. |
Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory. |
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Let <math>M</math> be a module over a ring <math>R</math> and <math>I</math> an ideal. Then <math>M</math> determines the sheaf <math>\widetilde{M}</math> on <math>Y = \operatorname{Spec}(R) - V(I)</math> (the restriction to ''Y'' of the sheaf associated to ''M''). Unwinding the definition, one sees: |
Let <math>M</math> be a module over a ring <math>R</math> and <math>I</math> an ideal. Then <math>M</math> determines the sheaf <math>\widetilde{M}</math> on <math>Y = \operatorname{Spec}(R) - V(I)</math> (the restriction to ''Y'' of the sheaf associated to ''M''). Unwinding the definition, one sees: |
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:<math>\Gamma_I(M) := \Gamma(Y, \widetilde{M}) = \varinjlim \operatorname{Hom}(I^n, M)</math>. |
:<math>\Gamma_I(M) := \Gamma(Y, \widetilde{M}) = \varinjlim \operatorname{Hom}(I^n, M)</math>. |
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Here, <math>\Gamma_I(M)</math> is called the '''ideal transform''' of <math>M</math> with respect to <math>I</math>. |
Here, <math>\Gamma_I(M)</math> is called the '''ideal transform''' of <math>M</math> with respect to <math>I</math>.{{CN|date=July 2023}} |
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== See also == |
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==References== |
==References== |
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*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}} |
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}} |
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* [[David Eisenbud|Eisenbud, David]], ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, {{ISBN|0-387-94268-8}}. |
* [[David Eisenbud|Eisenbud, David]], ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, {{ISBN|0-387-94268-8}}. |
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* {{Citation | last=Huneke | first=Craig | last2=Swanson | first2=Irena |author2-link= |
* {{Citation | last=Huneke | first=Craig | last2=Swanson | first2=Irena | author2-link=Irena Swanson | title=Integral closure of ideals, rings, and modules | url=http://people.reed.edu/~iswanson/book/index.html | publisher=[[Cambridge University Press]] | location=Cambridge, UK | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-68860-4 | mr=2266432 | year=2006 | volume=336 | access-date=2019-11-15 | archive-date=2019-11-15 | archive-url=https://web.archive.org/web/20191115053353/http://people.reed.edu/~iswanson/book/index.html | url-status=dead }} |
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{{DEFAULTSORT:Ideal Theory}} |
{{DEFAULTSORT:Ideal Theory}} |
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[[Category:Ideals]] |
[[Category:Ideals (ring theory)]] |
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[[Category:History of mathematics]] |
[[Category:History of mathematics]] |
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[[Category:Commutative algebra]] |
[[Category:Commutative algebra]] |
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{{abstract-algebra-stub}} |
Latest revision as of 22:00, 9 May 2024
In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.)
Throughout the articles, rings refer to commutative rings. See also the article ideal (ring theory) for basic operations such as sum or products of ideals.
Ideals in a finitely generated algebra over a field
[edit]This section needs expansion. You can help by adding to it. (May 2022) |
Ideals in a finitely generated algebra over a field (that is, a quotient of a polynomial ring over a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, if is a finitely generated algebra over a field, then the radical of an ideal in is the intersection of all maximal ideals containing the ideal (because is a Jacobson ring). This may be thought of as an extension of Hilbert's Nullstellensatz, which concerns the case when is a polynomial ring.
Topology determined by an ideal
[edit]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,
for some integer . This topology is called the I-adic topology. It is also called an a-adic topology if is generated by an element .
For example, take , the ring of integers and an ideal generated by a prime number p. For each integer , define when , prime to . Then, clearly,
where denotes an open ball of radius with center . Hence, the -adic topology on is the same as the metric space topology given by . As a metric space, can be completed. The resulting complete metric space has a structure of a ring that extended the ring structure of ; this ring is denoted as and is called the ring of p-adic integers.
Ideal class group
[edit]In a Dedekind domain 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 , an ideal is invertible in the sense: there exists a fractional ideal (that is, an A-submodule of ) such that , 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 ideal class group 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 integral domain, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki's Algèbre commutative gives such a theory.
The ideal class group of A, when it can be defined, is closely related to the Picard group of the spectrum of A (often the two are the same; e.g., for Dedekind domains).
In algebraic number theory, especially in class field theory, it is more convenient to use a generalization of an ideal class group called an idele class group.
Closure operations
[edit]This section needs expansion. You can help by adding to it. (May 2022) |
There are several operations on ideals that play roles of closures. The most basic one is the radical of an ideal. Another is the integral closure of an ideal. Given an irredundant primary decomposition , the intersection of 's whose radicals are minimal (don’t contain any of the radicals of other 's) is uniquely determined by ; this intersection is then called the unmixed part of . It is also a closure operation.
Given ideals in a ring , the ideal
is called the saturation of with respect to and is a closure operation (this notion is closely related to the study of local cohomology).
See also tight closure.
Reduction theory
[edit]Local cohomology in ideal theory
[edit]This section needs expansion. You can help by adding to it. (December 2019) |
Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory.
Let be a module over a ring and an ideal. Then determines the sheaf on (the restriction to Y of the sheaf associated to M). Unwinding the definition, one sees:
- .
Here, is called the ideal transform of with respect to .[citation needed]
See also
[edit]References
[edit]- Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
- Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.
- Huneke, Craig; Swanson, Irena (2006), Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432, archived from the original on 2019-11-15, retrieved 2019-11-15