Associated prime

In commutative algebra, a branch of mathematics, a prime ideal of a ring associated to a module if it is the annihilator of an element of.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. For more details see Commutative Algebra.

Definition

Is a prime ideal of and a module so called associated to, is if there is a so. So there is one in so applies to all:

The set of associated prime ideals is denoted by.

Sets

The following rates apply for a module over a ring:

• Is a submodule of, then

• Is not the zero module and noetherian, so is not empty.
• Is noetherian, then

• Is finite and finitely generated noetherian, then there exists a chain of submodules ( a composition series )

• General: Is noetherian and there is a composition series

• It follows in particular that a Noetherian ring has only finitely many minimal prime ideals contains.

Connection with the carrier

If a Noetherian ring and a module is equal to the zero module, then the support of the set of all prime ideals, the superset of an associated prime ideal is to have.