Associated prime ideal

In commutative algebra , a branch of mathematics, a prime ideal of a ring is associated to a module over when it is the annihilator of an element . ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle M}$

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 .

definition

Be a ring , be a prime ideal and be a - module . Then associated is said to be if one exists, such that: ${\displaystyle R}$ ${\displaystyle {\mathfrak {p}}\subseteq R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle M}$ ${\displaystyle m\in M}$

${\displaystyle {\mathfrak {p}}=\mathrm {Ann} (m)}$.

So there is an in , so that applies to all in : ${\displaystyle m}$ ${\displaystyle M}$ ${\displaystyle r}$ ${\displaystyle R}$

${\displaystyle r*m=0\Leftrightarrow r\in {\mathfrak {p}}}$

The set of associated prime ideals is denoted by. ${\displaystyle \mathrm {Ass} (M)}$

sentences

The following sentences apply to a module over a ring : ${\displaystyle M}$ ${\displaystyle R}$

• Is a sub-module of , so is ${\displaystyle U}$ ${\displaystyle M}$

${\displaystyle \mathrm {Ass} (U)\subset \mathrm {Ass} (M)\subset \mathrm {Ass} (U)\cup \mathrm {Ass} (M/U)}$

• If the null module is not and is noetherian , it is not empty. ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle \mathrm {Ass} (M)}$
• Is noetherian, so is ${\displaystyle R}$

${\displaystyle \bigcup _{{\mathfrak {p}}\in \mathrm {Ass} (M)}{\mathfrak {p}}}$

is the set of all zero divisors of . ${\displaystyle M}$

• If finitely generated and Noetherian, there is a chain of sub-modules (a series of compositions) ${\displaystyle M}$ ${\displaystyle R}$

${\displaystyle M=M_{0}\subset M_{1}\subset \dotsc \subset M_{n}=0}$

and a lot of prime ideals

${\displaystyle \{{\mathfrak {p}}_{i}|0\leq i<n,{\mathfrak {p}}\in \operatorname {Spec} (R)\}\supset \mathrm {Ass} (M)}$

so that is isomorphic to . In particular, in this case is a finite set. ${\displaystyle M_{i}/M_{i+1}}$ ${\displaystyle R/p_{i}}$ ${\displaystyle \mathrm {Ass} (M)}$

• General: Is noetherian and there is a series of compositions ${\displaystyle R}$

${\displaystyle M=M_{0}\subset M_{1}\subset \dotsc \subset M_{n}=0}$

so that is isomorphic to (with prime ideals ), then: ${\displaystyle M_{i}/M_{i+1}}$ ${\displaystyle R/{\mathfrak {p}}_{i}}$ ${\displaystyle {\mathfrak {p}}_{i}}$

${\displaystyle \mathrm {Ass} (M)\subset \{{\mathfrak {p}}_{0},\dotsc ,{\mathfrak {p}}_{n-1}\}\subset \mathrm {Supp} (M)}$

These three sets have the same minimal elements.

• From this it follows in particular that a Noetherian ring contains only a finite number of minimal prime ideals.

Connection with the carrier

If a Noetherian ring and a module are not equal to the zero module, then the carrier of the set of all prime ideals is the superset of a prime ideal to be associated. ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$

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 .
• Kunz: Introduction to Commutative Algebra and Algebraic Geometry. Vieweg, 1980, ISBN 3-528-07246-6 .
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94268-1 , MR 1322960
• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5 , MR 1653294
• Matsumura, Hideyuki (1970), Commutative algebra