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
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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 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
Ideal class group
In a Dedekind domain A (e.g., a ring of integers in a number field or the coordinate ring of a smooth affine curve), an ideal is invertible in the sense: there exists a fractional ideal such that , where the product on the left is a product of submodules of the field of fractions of A. In other words, ideals form a group under a product. The quotient of the group of 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.
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
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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 '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 Noetherian 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
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