Primary decomposition

The primary decomposition is a term used in commutative algebra. In a primary decomposition submodules are represented as the average primary submodules. Existence and uniqueness can be proved under certain conditions. The primary decomposition of an ideal is a generalization of the decomposition of a number into its prime numbers. On the other hand, the primary decomposition is the algebraic basis for the decomposition of an algebraic variety into its irreducible components.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. Ring homomorphisms form elements from one to one elements. For more details see Commutative Algebra.

Definition

A sub-module is a module over a ring, it is a primary separation of a representation of the average:

Of - primary submodules. ( Those are prime ideals of the ring. )

The primary decomposition is called reduced if the following holds:

At a reduced primary decomposition also referred to as the primary components.

Existence

If a finitely generated module over a ring noetherschem, so has any true submodule of a reduced primary decomposition. In particular, every ideal has as a sub- module of a decomposition into primary ideals.

Unambiguity

Is a submodule of a module over a Noetherian ring and

A reduced primary decomposition is in - primary submodules, so

Is the set of associated prime ideals of. In particular, the set of prime ideals occurring at a reduced primary decomposition is uniquely determined.

Is a minimal element of the set: so is the same. The minimal elements of corresponding primary components are defined by and unambiguous.

Heard a primary component not to a minimal element of such an embedded primary component is called. These are not necessarily unique ( see below).

Sets

Is a multiplicatively closed subset of a ring and

A reduced primary decomposition of a submodule with - primary submodules of, so is

A reduced primary representation of.

Examples

The integers

For example, in all the figures

With prime numbers, as is the primary decomposition of

In a coordinate ring

The ideal is a body that has such

The primary decompositions:

Primarily as a power of a maximal ideal; in the ring every zero divisor is nilpotent, so the ideal is primary. Both and are -primary.