Associated prime to a module: Difference between revisions

Latest revision as of 16:18, 12 May 2008

Definition

Let $R$ be a commutative unital ring and $M$ be a $R$ -module. A prime ideal $P$ of $R$ is said to be associated to $M$ if it satisfies the following equivalent conditions:

$P$ is the annihilator of an element of $M$
There is an injective homomorphism $A / P \to M$ of $A$ -modules

The set of all primes associated to $M$ is denoted as $A s s_{R} M$ .

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 ==Definition==
-Let <math>R</math> be a [[commutative unital ring]] and <math>M</math> be a <math>R</math>-module. A [[prime ideal]] <math>P</math> of <math>R</math> is said to be ''associated'' to <math>M</math> if <math>P</math> is the annihilator of an element of <math>M</math>.
+Let <math>R</math> be a [[commutative unital ring]] and <math>M</math> be a <math>R</math>-module. A [[prime ideal]] <math>P</math> of <math>R</math> is said to be ''associated'' to <math>M</math> if it satisfies the following equivalent conditions:
+* <math>P</math> is the annihilator of an element of <math>M</math>
+* There is an injective homomorphism <math>A/P \to M</math> of <math>A</math>-modules
 The set of all primes associated to <math>M</math> is denoted as <math>Ass_RM</math>.