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
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Ideals in a finitely generated algebra (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, 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
Closure operations
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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 , the intersection of whose radicals are minimal (not contained in the other radicals of primary ideals) is uniquely determined by ; this intersection is then called the unmixed part of . It is also a closure operation.
See also tight closure.
Reduction theory
Local cohomology in ideal theory
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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 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 .[1]
See also
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