Primary ideal: Difference between revisions

Latest revision as of 17:56, 18 December 2011

Definition

Equivalent definitions in tabular format

No.	Shorthand	An ideal in a commutative unital ring is termed a primary ideal if ...	An ideal $I$ in a commutative unital ring is termed a primary ideal if ...
1	product of two elements version	it is a proper ideal in the ring and whenever the product of two elements of the ring lies in the ideal, either the first element lies in the ideal or some power of the second element lies in the ideal.	$I$ is proper in $R$ and whenever $a,b\in R$ (possibly equal, possibly distinct) are such that $ab\in I$ , then either $a\in I$ or there exists a positive integer $n$ such that $b^{n}\in I$ .
2	exactly one associated prime	there is exactly one associated prime to the ideal	Fill this in later
3	quotient ring is primary	the quotient ring is a primary ring, i.e., it has exactly one associated prime.	the quotient ring $R/I$ is a primary ring, i.e., it has exactly one associated prime.

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: primary ring | View other quotient-determined properties of ideals in commutative unital rings

Examples

Important ring types

Property of commutative unital rings	Characterization of primary ideals in such a ring	Examples
principal ideal domain (i.e., an integral domain in which every ideal is a principal ideal)	either the zero ideal or a principal ideal generated by a nontrivial (i.e., use nonzero exponent) power of a prime element . In particular, for a PID, the primary ideals coincide with the powers of prime ideals, and also with the ideals with prime radical.	In the ring of integers, the primary ideals are precisely the zero ideal and the ideals generated by powers of primes. Thus, $3\mathbb {Z} ,4\mathbb {Z} ,5\mathbb {Z} ,8\mathbb {Z}$ are primary ideals. On the other hand, $6\mathbb {Z}$ and $24\mathbb {Z}$ are not primary. In the (univariate) polynomial ring over a field, the primary ideals are precisely the principal ideals generated by a power of an irreducible polynomial.
Dedekind domain (roughly speaking, unique factorization of ideals into prime ideals)	coincides with power of a prime ideal
unique factorization domain	the principal primary ideals are precisely the ideals generated by powers of primes. The non-principal primary ideals are harder to describe.	In the multivariate polynomial ring over a field, ...

Relation with other properties

@@ Line 1: / Line 1: @@
+==Definition==
+===Equivalent definitions in tabular format===
+{| class="sortable" border="1"
+! No. !! Shorthand !! An [[ideal]] in a [[commutative unital ring]] is termed a primary ideal if ... !! An [[ideal]] <math>I</math> in a [[commutative unital ring]] is termed a primary ideal if ...
+|-
+| 1 || product of two elements version || it is a [[proper ideal]] in the ring and whenever the product of two elements of the ring lies in the ideal, either the first element lies in the ideal or some power of the second element lies in the ideal. || <math>I</math> is proper in <math>R</math> and whenever <math>a,b \in R</math> (possibly equal, possibly distinct) are such that <math>ab \in I</math>, then either <math>a \in I</math> or there exists a positive integer <math>n</math> such that <math>b^n \in I</math>.
+|-
+| 2 || exactly one associated prime || there is exactly one [[associated prime to an ideal|associated prime to the ideal]] || {{fillin}}
+|-
+| 3 ||quotient ring is primary || the quotient ring is a [[primary ring]], i.e., it has exactly one associated prime. || the quotient ring <math>R/I</math> is a primary ring, i.e., it has exactly one associated prime.
+|}
+{{TOCright}}
 {{curing-ideal property}}
 {{quotient is a|primary ring}}
-==Definition==
+==Examples==
-===Symbol-free definition===
-An [[ideal]] in a [[commutative unital ring]] (or in any [[commutative ring]]) is termed '''primary''' if it satisfies the condition that whenever the product of two elements of the ring lies inside the ideal, either the first element lies inside the ideal or a suitable power of the second element lies inside the ideal.
-===Definition with symbols===
-An [[ideal]] <math>I</math> in a commutative ring <math>R</math> is termed '''primary''' if for any <math>p,q</math> in <math>R</math> such that <math>pq</math> is in <math>I</math>, either <math>p</math> is in <math>I</math>, or there exists a positive integer <math>n</math> such that <math>q^n</math> lies in <math>I</math>.
+===Important ring types===
+{| class="sortable" border="1"
+! Property of commutative unital rings !! Characterization of primary ideals in such a ring !! Examples
+|-
+| [[principal ideal domain]] (i.e., an [[integral domain]] in which every ideal is a [[principal ideal]]) || either the zero ideal or a principal ideal generated by a nontrivial (i.e., use nonzero exponent) power of a prime element . In particular, for a PID, the primary ideals coincide with the powers of prime ideals, and also with the [[ideal with prime radical|ideals with prime radical]]. || In the [[ring of integers]], the primary ideals are precisely the zero ideal and the ideals generated by powers of primes. Thus, <math>3\mathbb{Z}, 4 \mathbb{Z}, 5\mathbb{Z}, 8\mathbb{Z}</math> are primary ideals. On the other hand, <math>6\mathbb{Z}</math> and <math>24\mathbb{Z}</math> are not primary.<br>In the (univariate) [[polynomial ring over a field]], the primary ideals are precisely the principal ideals generated by a power of an irreducible polynomial.
+|-
+| [[Dedekind domain]] (roughly speaking, unique factorization of ideals into prime ideals) || coincides with [[power of a prime ideal]] ||
+|-
+| [[unique factorization domain]] || the ''principal'' primary ideals are precisely the ideals generated by powers of primes. The non-principal primary ideals are harder to describe. || In the [[multivariate polynomial ring over a field]], ...
+|}
 ==Relation with other properties==
 ===Stronger properties===
-* [[Maximal ideal]]
+* [[Weaker than::Maximal ideal]]
-* [[Prime ideal]]
+* [[Weaker than::Prime ideal]]
-* [[Ideal with maximal radical]]
+* [[Weaker than::Ideal with maximal radical]]
-* [[Irreducible ideal]] if the ring is [[Noetherian ring|Noetherian]]: {{proofat|[[Irreducible implies primary (Noetherian)]]
+* [[Irreducible ideal]] if the ring is [[Noetherian ring|Noetherian]]: {{proofat|[[Irreducible implies primary (Noetherian)]]}}
 ===Weaker properties===
-* [[Ideal with prime radical]]
+* [[Stronger than::Ideal with prime radical]]
 ===Incomparable properties===
 * [[Radical ideal]]