Blueprint for the Adjunction Formula

1.8 Associated Primes

Proposition 1.8.1
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Let \(R\) a ring, and \(M\) an \(R\)-module. Let \(\mathfrak {p}\) be a prime ideal, then the following are equivalent:

  1. \(\mathfrak {p} = \operatorname {Ann}_R (m)\) for some element \(m \in M\).

  2. \(R / \mathfrak {p}\) embeds into \(M\).

Definition 1.8.2

The set \(\operatorname {Assoc}_R (M)\) is the set of primes satisfying the preivious proposition

Proposition 1.8.3
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The associated primes of \(M\) are in the support of \(M\).

Lemma 1.8.4

The assocated primes of \(M\) are precisely the primes which are minimal in the support of \(M\).

Lemma 1.8.5

An element \(x \in R\) is a zero-divisor iff it is in an associated prime. (geometrically, it vanishes at an associated point)