Ideal theory

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

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 $A$ is a finitely generated algebra over a field, then the radical of an ideal in $A$ is the intersection of all maximal ideals containing the ideal (because $A$ is a Jacobson ring). This may be thought of as an extension of Hilbert's Nullstellensatz, which concerns the case when $A$ is a polynomial ring.

Topology determined by an ideal

Closure operations

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 $I=\cap Q_{i}$ , the intersection of $Q_{i}$ 's whose radicals are minimal (don’t contain any of the radicals of other $Q_{i}$ ) is uniquely determined by $I$ ; this intersection is then called the unmixed part of $I$ . It is also a closure operation.

Given ideals $I,J$ in a Noetherian ring $A$ , the ideal

(I:J^{\infty })=\{f\in A|fJ^{n}\subset I,n\gg 0\}

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

Reduction theory

Local cohomology in ideal theory

Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory.

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

\Gamma _{I}(M):=\Gamma (Y,{\widetilde {M}})=\varinjlim \operatorname {Hom} (I^{n},M)

.

Here, $\Gamma _{I}(M)$ is called the ideal transform of $M$ with respect to $I$ .^[1]

References

^ Eisenbud 2005, Appendix 10B.

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

[1] Eisenbud 2005, Appendix 10B.

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