Primary ideal

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

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .

Definitions

Primary module

A sub- module of a module over a ring is a primary sub-module if it has only one associated prime ideal . This is equivalent to saying that for all of them the figure: ${\displaystyle U}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle r\in R}$

${\displaystyle h_{r}\colon M\to M}$

${\displaystyle h_{r}\colon m\mapsto r*m}$

is either injective or nilpotent .

If the associated prime ideal is also referred to as -primary sub-module. ${\displaystyle p}$ ${\displaystyle U}$ ${\displaystyle p}$

Primary ideal

A ring ideal is a primary ideal when it is a sub-module of a primary sub-module. This is equivalent to having every zero divisor of nilpotent. ${\displaystyle p}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R/p}$

Expressed in terms of elements , this means: An ideal is primary if . ${\displaystyle R}$ ${\displaystyle p\subset R}$ ${\displaystyle \forall r,s\in R:(rs\in p\rightarrow r\in p{\text{ oder }}\exists m>0:s^{m}\in p)}$

properties

If there is a module, then: ${\displaystyle M}$ ${\displaystyle R}$

• Every prime ideal is a primary ideal.
• If an ideal is -primär, then there is one , so is, ${\displaystyle q\ p}$ ${\displaystyle n}$ ${\displaystyle p^{n}\subset q\subset p}$
• The reverse of the last sentence is wrong. But if there is a maximum ideal of a Noetherian ring, then an ideal is -primary if and only if there is one such that is. ${\displaystyle m}$ ${\displaystyle q}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle m^{n}\subset q\subset m}$
• If is Noether , then the intersection of finitely many -primary sub- modules of -primary. ${\displaystyle R}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle p}$
• If is noetherian and is an irreducible proper sub-module of , then is primary. ${\displaystyle R}$ ${\displaystyle Q}$ ${\displaystyle R}$ ${\displaystyle Q}$

literature

• Atiyah, Macdonald: Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6