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| In [[abstract algebra]], an '''associated prime''' of a [[module (mathematics)|module]] ''M'' over a [[ring (mathematics)|ring]] ''R'' is a type of [[prime ideal]] of ''R'' that arises as an [[annihilator (ring theory)|annihilator]] of a submodule of ''M''. The set of associated primes is usually denoted by <math>\operatorname{Ass}_R(M)\,</math>.
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| In [[commutative algebra]], associated primes are linked to the [[Lasker–Noether theorem|Lasker-Noether primary decomposition]] of ideals in commutative [[Noetherian ring]]s. Specifically, if an ideal ''J'' is decomposed as a finite intersection of [[primary ideal]]s, the [[radical of an ideal|radicals]] of these primary ideals are [[prime ideal]]s, and this set of prime ideals coincides with <math>\operatorname{Ass}_R(R/J)\,</math>.{{sfn|Lam|1999|p=117|loc=Ex 40B}} Also linked with the concept of "associated primes" of the ideal are the notions of '''isolated primes''' and '''embedded primes'''.
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| ==Definitions==
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| A nonzero ''R'' module ''N'' is called a '''prime module''' if the annihilator <math>\mathrm{Ann}_R(N)=\mathrm{Ann}_R(N')\,</math> for any nonzero submodule ''N' '' of ''N''. For a prime module ''N'', <math>\mathrm{Ann}_R(N)\,</math> is a prime ideal in ''R''.{{sfn|Lam|1999|p=85}}
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| An '''associated prime''' of an ''R'' module ''M'' is an ideal of the form <math>\mathrm{Ann}_R(N)\,</math> where ''N'' is a prime submodule of ''M''. In commutative algebra the usual definition is different, but equivalent:{{sfn|Lam|1999|p=86}} if ''R'' is commutative, an associated prime ''P'' of ''M'' is a prime ideal of the form <math>\mathrm{Ann}_R(m)\,</math> for a nonzero element ''m'' of ''M'' or equivalently <math>R/P</math> is isomorphic to a submodule of ''M''.
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| In a commutative ring ''R'', minimal elements in <math>\operatorname{Ass}_R(M)</math> (with respect to the set-theoretic inclusion) are called '''isolated primes''' while the rest of the associated primes (i.e., those properly containing associated primes) are called '''embedded prime'''.
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| A module is called '''coprimary''' if ''xm'' = 0 for some nonzero ''m'' ∈ ''M'' implies ''x''<sup>''n''</sup>''M'' = 0 for some positive integer ''n''. A nonzero finitely generated module ''M'' over a commutative [[Noetherian ring]] is coprimary if and only if it has exactly one associated prime. A submodule ''N'' of ''M'' is called ''P''-primary if <math>M/N</math> is coprimary with ''P''. An ideal ''I'' is a ''P''-[[primary ideal]] if and only if <math>\operatorname{Ass}_R(R/I) = \{P\}</math>; thus, the notion is a generalization of a primary ideal.
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| ==Properties==
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| Most of these properties and assertions are given in {{harv|Lam|2001}} starting on page 86.
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| * If ''M' ''⊆''M'', then <math>\mathrm{Ass}_R(M')\subseteq\mathrm{Ass}_R(M)\,</math>. If in addition ''M' '' is an [[essential submodule]] of ''M'', their associated primes coincide.
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| * It is possible, even for a commutative local ring, that the set of associated primes of a [[finitely generated module]] is empty. However, in any ring satisfying the [[ascending chain condition]] on ideals (for example, any right or left Noetherian ring) every nonzero module has at least one associated prime.
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| * Any [[uniform module]] has either zero or one associated primes, making uniform modules an example of coprimary modules.
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| * For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable [[injective module]]s onto the [[spectrum of a ring|spectrum]] <math>\mathrm{Spec}(R)\,</math>. If ''R'' is an [[Artinian ring]], then this map becomes a bijection.
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| *'''Matlis' Theorem''': For a commutative Noetherian ring ''R'', the map from the isomorphism classes of indecomposable injective modules to the spectrum is a bijection. Moreover, a complete set of representatives for those classes is given by <math>E(R/\mathfrak{p})\,</math> where <math>E(-)\,</math> denotes the [[injective hull]] and <math>\mathfrak{p}\,</math> ranges over the prime ideals of ''R''.
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| * For a [[Noetherian module]] ''M'' over any ring, there are only finitely many associated primes of ''M''.
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| The following properties all refer to a commutative Noetherian ring ''R'': | |
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| * Every ideal ''J'' (through primary decomposition) is expressible as a finite intersection of primary ideals. The radical of each of these ideals is a prime ideal, and these primes are exactly the elements of <math>\mathrm{Ass}_R(R/J)\,</math>. In particular, an ideal ''J'' is a [[primary ideal]] if and only if <math>\mathrm{Ass}_R(R/J)\,</math> has exactly one element.
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| * Any [[minimal prime ideal|prime ideal minimal]] with respect to containing an ideal ''J'' is in <math>\mathrm{Ass}_R(R/J)\,</math>. These primes are precisely the isolated primes.
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| * The set theoretic union of the associated primes of ''M'' is exactly the collection of zero-divisors on ''M'', that is, elements ''r'' for which there exists nonzero ''m'' in ''M'' with ''mr'' =0.
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| *If ''M'' is a finitely generated module over ''R'', then there is a finite ascending sequence of submodules
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| :: <math>0=M_0\subset M_1\subset\cdots\subset M_{n-1}\subset M_n=M\,</math>
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| :such that each quotient ''M''<sub>''i''</sub>/''M''<sub>''i−1''</sub> is isomorphic to ''R''/''P''<sub>''i''</sub> for some prime ideals ''P''<sub>''i''</sub>. Moreover every associated prime of ''M'' occurs among the set of primes ''P''<sub>''i''</sub>. (In general not all the ideals ''P''<sub>''i''</sub> are associated primes of ''M''.)
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| * Let ''S'' be a multiplicatively closed subset of ''R'' and <math>f: \operatorname{Spec}(S^{-1}R) \to \operatorname{Spec}(R)</math> the canonical map. Then, for a module ''M'' over ''R'',
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| *:<math>\operatorname{Ass}_R(S^{-1}M) = f(\operatorname{Ass}_{S^{-1}R}(S^{-1}M)) = \operatorname{Ass}_R(M) \cap \{ P | P \cap S = \emptyset \}</math>.<ref>{{harvnb|Matsumura|1970|loc=7.C Lemma}}</ref>
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| * For a module ''M'' over ''R'', <math>\mathrm{Ass}(M) \subseteq \mathrm{Supp}(M)</math>. Furthermore, the set of minimal elements of <math>\mathrm{Supp}(M)</math> coincides with the set of minimal elements of <math>\mathrm{Ass}(M)</math>. In particular, the equality holds if <math>\mathrm{Ass}(M)</math> consists of maximal ideals.
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| * A module ''M'' over ''R'' has [[finite length]] if and only if ''M'' is finitely generated and <math>\mathrm{Ass}(M)</math> consists of maximal ideals.{{citation needed|date=August 2013}}<!-- or just add a proof? -->
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| ==Examples==
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| *If ''R'' is the ring of integers, then non-trivial [[free abelian group]]s and non-trivial [[abelian group]]s of prime power order are coprimary.
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| *If ''R'' is the ring of integers and ''M'' a finite abelian group, then the associated primes of ''M'' are exactly the primes dividing the order of ''M''.
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| *The group of order 2 is a quotient of the integers ''Z'' (considered as a free module over itself), but its associated prime ideal (2) is not an associated prime of ''Z''.
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| ==References==
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| {{Reflist}}
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| *{{Citation | last1=Eisenbud | first1=David | author1-link=David Eisenbud | title=Commutative algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-94268-1 | mr=1322960 | year=1995 | volume=150}}
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| *{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999}}
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| * {{citation | last1=Matsumura |first1=Hideyuki |title=Commutative algebra |year=1970}}
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| [[Category:Commutative algebra]]
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| [[Category:Ideals]]
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| [[Category:Prime ideals]]
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| [[Category:Module theory]]
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