Prime ideal: Difference between revisions

Latest revision as of 16:57, 18 December 2011

Definition

Equivalent definitions in tabular format

No.	Shorthand	An ideal in a commutative unital ring is termed a prime ideal if ...	An ideal $I$ in a commutative unital ring $R$ is termed a prime ideal if ...
1	product of two elements version	It is a proper ideal and whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal	$I$ is proper in $R$ and whenever $a,b\in R$ are (possibly equal, possibly distinct elements) such that $ab\in I$ then either $a\in I$ or $b\in I$ (note: it's also possible that both $a,b$ are in $I$ ).
2	product of two ideals version	It is a proper ideal and whenever the product of two ideals is contained in it, one of the ideals is contained in it.	$I$ is proper in $R$ and whenever $J$ and $K$ are (possibly equal, possibly distinct) ideals such that $JK\subseteq I$ , then either $J\subseteq I$ or $K\subseteq I$ (note: it's also possible that both $J,K$ are in $I$ ).
3	product of finitely many elements version	It is a proper ideal and whenever the product of finitely many elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal	$I$ is proper in $R$ and whenever $a_{1},a_{2},\ldots ,a_{n}\in R$ are such that $a_{1}a_{2}\ldots a_{n}\in I$ , there exists $i\in \{1,2,\ldots ,n\}$ such that $a_{i}\in I$ . Note that the $a_{i}$ may be equal or distinct, i.e., some may be equal, some may be distinct.
4	product of finitely many ideals version	It is a proper ideal and whenever the product of finitely many ideals is contained in it, one of the ideals is contained in it.	$I$ is proper in $R$ and whenever $J_{1},J_{2},\ldots ,J_{n}$ are ideals in $R$ such that $J_{1}J_{2}\ldots J_{n}\subseteq I$ , there exists $i\in \{1,2,\ldots ,n\}$ such that $J_{i}\subseteq I$
5	complement saturated subset version	It is a proper ideal and its complement is a multiplicatively closed saturated subset (that is, is closed with respect to the operations of multiplication and factorization).	$I$ is proper in $R$ and the complement of $I$ in $R$ is a multiplicatively closed and saturated subset i.e. $ab\in R\setminus I\iff a\in R\setminus I$ or $b\in R\setminus I$ .
6	quotient is integral domain	The quotient ring is an integral domain, i.e., a nonzero ring where the product of any two nonzero elements is nonzero.	The quotient ring $R/I$ is an integral domain.

The set of prime ideals

Further information: Spectrum of a commutative unital ring

The set of prime ideals in a commutative unital ring is termed its spectrum. The spectrum is more than just a set, it has the structure of a topological space. In fact, it is a locally ringed space.

This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra

This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings

This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: integral domain | View other quotient-determined properties of ideals in commutative unital rings

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Intersection-closedness

This property of ideals in commutative unital rings is not closed under taking arbitrary intersections; in other words, an arbitrary intersection of ideals with this property need not have this property

An intersection of prime ideals need not be prime. In fact, an ideal is an intersection of prime ideals iff it is a radical ideal.

Contraction-closedness

This property of ideals in commutative unital rings is contraction-closed: a contraction of an ideal with this property, also has this property
View other contraction-closed properties of ideals in commutative unital rings

If $f:R\to S$ is a homomorphism of commutative unital rings, and $I$ is a prime ideal of $S$ , then the contraction of $I$ to $R$ , denoted $I^{c}$ , (same as $f^{-1}(I)$ ) is a prime ideal in $R$ .

For full proof, refer: primeness is contraction-closed

Note that this implies the intermediate subring condition and the transfer condition on ideals.

Intermediate subring condition

This property of ideals satisfies the intermediate subring condition for ideals: if an ideal has this property in the whole ring, it also has this property in any intermediate subring
View other properties of ideals satisfying the intermediate subring condition

If an ideal is prime in the whole ring, it is also prime in any intermediate subring. This is related to the fact that any subring of an integral domain is an integral domain.

Transfer condition

This property of ideals satisfies the transfer condition for ideals: if an ideal satisfies the property in the ring, its intersection with any subring satisfies the property inside that subring

If $I$ is a prime ideal in $R$ , and $S$ is any subring of $R$ , then $I\cap S$ is a prime ideal in $S$ . Note that this implies the intermediate ring condition as well.

Effect of property operators

The intersection-closure

Applying the intersection-closure to this property gives: radical ideal

An ideal in a commutative unital ring is expressible as an intersection of prime ideals iff it is a radical ideal.

Further information: Intersection of prime equals radical

In particular kinds of rings

In rings of integers

Further information: Prime ideal in ring of integers

A ring of integers in a number field is a Dedekind domain, hence any nonzero prime ideal in this ring is a maximal ideal.

In affine rings

Further information: Prime ideal in affine ring

Ring property assumed	Nature of prime ideals
Zero-dimensional ring	maximal
One-dimensional domain	Zero or maximal
Principal ideal domain	Zero or maximal
Dedekind domain	Zero or maximal

References

Book:Eisenbud, Page 12

External links

Search for "prime+ideal" on the World Wide Web:
Scholarly articles: Google Scholar, JSTOR
Books: Google Books, Amazon
This wiki: Internal search, Google site search
Encyclopaedias: Wikipedia (or using Google), Citizendium
Math resource pages:Mathworld, Planetmath, Springer Online Reference Works
Math wikis: Topospaces, Diffgeom, Commalg, Noncommutative
Discussion fora: Mathlinks, Google Groups
The web: Google, Yahoo

Definition links

@@ Line 1: / Line 1: @@
-==Definition for commutative rings==
+==Definition==
-===Symbol-free definition===
+===Equivalent definitions in tabular format===
-An [[ideal]] in a [[commutative unital ring]] (or in any [[commutative ring]]) is termed a '''prime ideal''' if it satisfies the following equivalent conditions:
+{| class="sortable" border="1"
+! No. !! Shorthand !! An [[ideal]] in a [[commutative unital ring]] is termed a [[prime ideal]] if ... !! An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is termed a [[prime ideal]] if ...
+|-
+| 1 || product of two elements version || It is a [[proper ideal]] and whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal || <math>I</math> is proper in <math>R</math> and whenever <math>a, b \in R</math> are (possibly equal, possibly distinct elements) such that <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math> (note: it's also possible that both <math>a,b</math> are in <math>I</math>).
+|-
+| 2 || product of two ideals version ||  It is a [[proper ideal]] and whenever the [[product of ideals|product]] of two ideals is contained in it, one of the ideals is contained in it. || <math>I</math> is proper in <math>R</math> and whenever <math>J</math> and <math>K</math> are (possibly equal, possibly distinct) ideals such that <math>JK \subseteq I</math>, then either <math>J \subseteq I</math> or <math>K \subseteq I</math> (note: it's also possible that both <math>J,K</math> are in <math>I</math>).
+|-
+| 3 || product of finitely many elements version || It is a [[proper ideal]] and whenever the product of finitely many elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal || <math>I</math> is proper in <math>R</math> and whenever <math>a_1,a_2,\ldots,a_n \in R</math> are such that <math>a_1a_2 \ldots a_n \in I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>a_i \in I</math>. Note that the <math>a_i</math> may be equal or distinct, i.e., some may be equal, some may be distinct.
+|-
+| 4 || product of finitely many ideals version || It is a [[proper ideal]] and whenever the product of finitely many ideals is contained in it, one of the ideals is contained in it. || <math>I</math> is proper in <math>R</math> and whenever <math>J_1,J_2,\ldots,J_n</math> are [[ideal]]s in <math>R</math> such that <math>J_1J_2\ldots J_n \subseteq I</math>, there exists <math>i \in \{ 1,2,\ldots, n\}</math> such that <math>J_i \subseteq I</math>
+|-
+| 5 || complement saturated subset version || It is a [[proper ideal]] and its complement is a multiplicatively closed [[defining ingredient::saturated subset]] (that is, is closed with respect to the operations of multiplication and factorization). || <math>I</math> is proper in <math>R</math> and the complement of <math>I</math> in <math>R</math> is a multiplicatively closed and saturated subset i.e. <math>ab \in R \setminus I \iff a \in R \setminus I</math> or <math>b \in R \setminus I</math>.
+|-
+| 6 || quotient is integral domain || The [[defining ingredient::quotient ring]] is an [[defining ingredient::integral domain]], i.e., a nonzero ring where the product of any two nonzero elements is nonzero. || The quotient ring <math>R/I</math> is an integral domain.
+|}
-* Whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal.
+===The set of prime ideals===
-* It is an ideal whose complement is a [[saturated subset]] (that is, is clsoed with respect to the operation of multiplication).
-* The [[quotient ring]] by that ideal is an [[integral domain]]
-===Definition with symbols===
+{{further|[[Spectrum of a commutative unital ring]]}}
-An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is termed a '''prime ideal''' if whenever <math>a, b \in R</math> are such that <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math>.
+The set of prime ideals in a commutative unital ring is termed its [[spectrum]]. The spectrum is more than just a set, it has the structure of a topological space. In fact, it is a locally ringed space.
-==Definition for non-commutative rings==
+{{basicdef}}
+{{curing-ideal property}}
+{{quotient is a|integral domain}}
-The definition has many different forms for noncommutative rings:
+==Relation with other properties==
-* [[Prime ideal (noncommutative rings)]]
+===Stronger properties===
-* [[Completely prime ideal]]
-[[Category: Properties of ideals in commutative rings]]
+* [[Weaker than::Maximal ideal]]
-[[Category: Quotient-determined properties of ideals in commutative rings]]
+* [[Weaker than::Minimal prime ideal]]
+* [[Weaker than::Principal prime ideal]]
+===Weaker properties===
+* [[Stronger than::Irreducible ideal]]
+* [[Stronger than::Primary ideal]]
+* [[Stronger than::Radical ideal]]
+* [[Stronger than::Ideal with prime radical]]
+==Metaproperties==
+{{not intersection-closed ideal property}}
+An intersection of prime ideals need not be prime. In fact, an ideal is an intersection of prime ideals iff it is a [[radical ideal]].
+{{contraction-closed ideal property}}
+If <math>f:R \to S</math> is a [[homomorphism of commutative unital rings]], and <math>I</math> is a [[prime ideal]] of <math>S</math>, then the [[contraction]] of <math>I</math> to <math>R</math>, denoted <math>I^c</math>, (same as <math>f^{-1}(I)</math>) is a prime ideal in <math>R</math>.
+{{proofat|[[primeness is contraction-closed]]}}
+Note that this implies the intermediate subring condition and the transfer condition on ideals.
+{{intringcondn ideal}}
+If an ideal is prime in the whole ring, it is also prime in any intermediate subring. This is related to the fact that any subring of an integral domain is an integral domain.
+{{transfercondn ideal}}
+If <math>I</math> is a prime ideal in <math>R</math>, and <math>S</math> is any subring of <math>R</math>, then <math>I \cap S</math> is a prime ideal in <math>S</math>. Note that this implies the intermediate ring condition as well.
+==Effect of property operators==
+{{applyingoperatorgives|intersection-closure|radical ideal}}
+An ideal in a commutative unital ring is expressible as an intersection of prime ideals iff it is a [[radical ideal]].
+{{further|[[Intersection of prime equals radical]]}}
+==In particular kinds of rings==
+===In rings of integers===
+{{further|[[Prime ideal in ring of integers]]}}
+A [[ring of integers in a number field]] is a [[Dedekind domain]], hence any nonzero prime ideal in this ring is a [[maximal ideal]].
+===In affine rings===
+{{further|[[Prime ideal in affine ring]]}}
+{| class="wikitable" border="1"
+! Ring property assumed !! Nature of prime ideals
+|-
+|[[Zero-dimensional ring]] || [[Maximal ideal|maximal]]
+|-
+|[[One-dimensional domain]] || Zero or [[maximal ideal|maximal]]
+|-
+|[[Principal ideal domain]]|| Zero or [[maximal ideal|maximal]]
+|-
+|[[Dedekind domain]]||Zero or [[maximal ideal|maximal]]
+|}
+==References==
+* {{booklink|Eisenbud}}, Page 12
+==External links==
+{{searchbox|"prime+ideal"}}
+===Definition links===
+* {{wp|Prime ideal}}
+* {{mathworld|PrimeIdeal}}
+* {{planetmath|PrimeIdeal}}