Inmathematics,ideal theory is the theory ofideals incommutative rings. While the notion of an ideal exists also fornon-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 articleideal (ring theory) for basic operations such as sum or products of ideals.

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Ideals in a finitely generated algebra over a field (that is, a quotient of apolynomial ring over a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, if${\displaystyle A}$ is a finitely generated algebra over a field, then theradical of an ideal in${\displaystyle A}$ is the intersection of all maximal ideals containing the ideal (because${\displaystyle A}$ is aJacobson ring). This may be thought of as an extension ofHilbert's Nullstellensatz, which concerns the case when${\displaystyle A}$ is a polynomial ring.

IfI is an ideal in a ringA, then it determines the topology onA where a subsetU ofA is open if, for eachx inU,

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

for some integer${\displaystyle n>0}$. This topology is called theI-adic topology. It is also called ana-adic topology if${\displaystyle I=aA}$ is generated by an element${\displaystyle a}$.

For example, take${\displaystyle A=\mathbb{Z}}$, the ring of integers and${\displaystyle I=pA}$ an ideal generated by a prime numberp. For each integer${\displaystyle x}$, define${\displaystyle |x{|}_p=p^{-n}}$ when${\displaystyle x=p^{n}y}$,${\displaystyle y}$prime to${\displaystyle p}$. Then, clearly,

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

where denotes an open ball of radius${\displaystyle r}$ with center${\displaystyle x}$. Hence, the${\displaystyle p}$-adic topology on${\displaystyle \mathbb{Z}}$ is the same as themetric space topology given by${\displaystyle d(x,y)=|x-y{|}_p}$. As a metric space,${\displaystyle \mathbb{Z}}$ can becompleted. The resulting complete metric space has a structure of a ring that extended the ring structure of${\displaystyle \mathbb{Z}}$; this ring is denoted as${\displaystyle {\mathbb{Z}}_p}$ and is called thering ofp-adic integers.

In aDedekind domainA (e.g., a ring of integers in a number field or the coordinate ring of a smooth affine curve) with the field of fractions${\displaystyle K}$, an ideal${\displaystyle I}$ is invertible in the sense: there exists afractional ideal${\displaystyle I^{-1}}$ (that is, anA-submodule of${\displaystyle K}$) such that${\displaystyle I{I}^{-1}=A}$, where the product on the left is a product of submodules ofK. 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 theideal class group ofA.

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 Noetherianintegral domain, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki'sAlgèbre commutative gives such a theory.

The ideal class group ofA, when it can be defined, is closely related to thePicard group of thespectrum ofA (often the two are the same; e.g., for Dedekind domains).

Inalgebraic number theory, especially inclass field theory, it is more convenient to use a generalization of an ideal class group called anidele class group.

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There are several operations on ideals that play roles of closures. The most basic one is theradical of an ideal. Another is theintegral closure of an ideal. Given an irredundant primary decomposition${\displaystyle I=\cap Q_i}$, the intersection of${\displaystyle Q_i}$'s whose radicals are minimal (don’t contain any of the radicals of other${\displaystyle Q_j}$'s) is uniquely determined by${\displaystyle I}$; this intersection is then called the unmixed part of${\displaystyle I}$. It is also a closure operation.

Given ideals${\displaystyle I,J}$ in a ring${\displaystyle A}$, the ideal

${\displaystyle (I:J^{\mathrm{\infty }})=\{f\in A\mid f{J}^n\subset I,n\gg 0\}=\bigcup _{n>0}{\mathrm{Ann}}_A((J^n+I)/I)}$

is called the saturation of${\displaystyle I}$ with respect to${\displaystyle J}$ and is a closure operation (this notion is closely related to the study of local cohomology).

See alsotight closure.

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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.

Let${\displaystyle M}$ be a module over a ring${\displaystyle R}$ and${\displaystyle I}$ an ideal. Then${\displaystyle M}$ determines the sheaf${\displaystyle \tilde{M}}$ on${\displaystyle Y=\mathrm{Spec}(R)-V(I)}$ (the restriction toY of the sheaf associated toM). Unwinding the definition, one sees:

${\displaystyle {\mathrm{\Gamma }}_I(M):=\mathrm{\Gamma }(Y,\tilde{M})=\underrightarrow{\lim }\mathrm{Hom}(I^n,M)}$.

Here,${\displaystyle {\mathrm{\Gamma }}_I(M)}$ is called theideal transform of${\displaystyle M}$ with respect to${\displaystyle I}$.^{[citation needed]}

• System of parameters

• 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 fromthe original on 2019-11-15, retrieved2019-11-15

• Ideals (ring theory)
• History of mathematics
• Commutative algebra

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