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
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
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...
, which concerns the case when
is a polynomial ring.
Topology determined by an ideal
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'',
:
for some integer
. This topology is called the ''I''-adic topology. It is also called an ''a''-adic topology if
is generated by an element
.
For example, take
, the ring of integers and
an ideal generated by a prime number ''p''. For each integer
, define
when
,
prime to . Then, clearly,
:
where
denotes an open ball of radius
with center
. Hence, the
-adic topology on
is the same as the
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
topology given by
. As a metric space,
can be
completed. The resulting complete metric space has a structure of a ring that extended the ring structure of
; this ring is denoted as
and is called the
ring of ''p''-adic integers.
Ideal class group
In a
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
''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
, an ideal
is invertible in the sense: there exists a
fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...
(that is, an ''A''-submodule of
) such that
, where the product on the left is a product of submodules of ''K''. 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 the
ideal class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a ...
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.
The ideal class group of ''A'', when it can be defined, is closely related to the
Picard group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a globa ...
of the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
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 In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the ...
.
Closure operations
There are several operations on ideals that play roles of closures. The most basic one is the
radical of an ideal
In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is calle ...
. 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 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
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
.
See also
*
System of parameters
In mathematics, a system of parameters for a local Noetherian ring of Krull dimension ''d'' with maximal ideal ''m'' is a set of elements ''x''1, ..., ''x'd'' that satisfies any of the following equivalent conditions:
# ''m'' is a minimal prim ...
References
*
*
Eisenbud, David, ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, .
*
{{DEFAULTSORT:Ideal Theory
Ideals (ring theory)
History of mathematics
Commutative algebra