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.
See also: finitely generated algebra.
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
A
A
A
See main article: I-adic topology.
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,
x+In\subsetU.
n>0
I=aA
a
For example, take
A=Z
I=pA
x
|x|p=p-n
x=pny
y
p
x+pnA=B(x,p-(n-1))
B(x,r)=\{z\inZ\mid|z-x|p<r\}
r
x
p
Z
d(x,y)=|x-y|p
Z
Z
Zp
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
K
I
I-1
K
II-1=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.
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=\capQi
Qi
Qj
I
I
Given ideals
I,J
A
(I:Jinfty)=\{f\inA\midfJn\subsetI,n\gg0\}=cupn
n | |
\operatorname{Ann} | |
A((J |
+I)/I)
I
J
See also tight closure.
See main article: Ideal reduction.
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
R
I
M
\widetilde{M}
Y=\operatorname{Spec}(R)-V(I)
\GammaI(M):=\Gamma(Y,\widetilde{M})=\varinjlim\operatorname{Hom}(In,M)
\GammaI(M)
M
I