Primary ideal

In commutative algebra, a primary ideal or primary ideal is a generalization of a prime power, just like a prime ideal is a generalization of a prime number. Primary ideals play an important role in the primary decomposition of modules.

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

Definitions

The primary module

A submodule of a module over a ring is a primary sub-module, when it has only one associated prime ideal. This is equivalent to the fact that all the picture:

Is either injective or nilpotent.

If the associated prime ideal, it is also referred to as the - primary submodule.

Primary Ideal

An ideal of a ring is a primary ideal if it is a submodule of a primary submodule. This is equivalent to saying that every zero divisor of is nilpotent.

Properties

If a module, then:

  • Every prime ideal is a primary ideal.
  • If an ideal -primary, then there is a, so is.
  • The reversal of the previous sentence is false. An ideal but is a maximal ideal of a Noetherian ring, clearly then - primary if there is one, so is.
  • When is noetherian, so the average is finitely many - primary submodules of -primary.
  • If is Noetherian and is a real irreducible submodule of, then is primary.
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