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.)

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 (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

This may be thought of as an extension of Hilbert's Nullstellensatz, which concerns the case when

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

It is also called an a-adic topology if

an ideal generated by a prime number p. For each integer

denotes an open ball of radius

is the same as the metric space topology given by

The resulting complete metric space has a structure of a ring that extended the ring structure of

and is called the ring of p-adic integers.

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

is invertible in the sense: there exists a fractional ideal

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.

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

; this intersection is then called the unmixed part of

, the ideal is called the saturation of

and is a closure operation (this notion is closely related to the study of local cohomology).