Primary ideal: Difference between revisions

Revision as of 17:21, 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

Relation with other properties

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-{{curing-ideal property}}
-{{quotient is a|primary ring}}
 ==Definition==
-===Symbol-free definition===
-An [[ideal]] in a [[commutative unital ring]] is termed '''primary''' if it satisfies the following equivalent conditions:
+===Equivalent definitions in tabular format===
-* 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
-* There is exactly one [[associated prime to an ideal|associated prime to the ideal]], i.e. exactly one [[associated prime to a module|associated prime]] to the quotient ring
-===Definition with symbols===
+{| 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.
+|}
-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>.
+{{curing-ideal property}}
+{{quotient is a|primary ring}}
 ==Relation with other properties==

Revision as of 17:21, 18 December 2011

Definition

Equivalent definitions in tabular format

Relation with other properties

Stronger properties

Weaker properties

Incomparable properties