Current (mathematics)

2013-12-25T11:55:32Z

94.68.199.36: /* Homological theory */ reworded the use of stokes' theorem

In [[mathematics]], specifically [[commutative algebra]], a proper [[Ideal (ring theory)|ideal]] ''Q'' of a [[commutative ring]] ''A'' is said to be '''primary''' if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y''''n'' is also an element of ''Q'', for some ''n>0''. For example, in the [[ring of integers]] '''Z''', (''p''n) is a primary ideal if ''p'' is a prime number.

The notion of primary ideals is important in commutative ring theory because every ideal of a [[Noetherian ring]] has a [[primary decomposition]], that is, can be written as an intersection of finitely many primary ideals. This result is known as the [[Lasker–Noether theorem]]. Consequently,<ref>To be precise, one usually uses this fact to prove the theorem.</ref> an [[irreducible ideal]] of a Noetherian ring is primary.

Various methods of generalizing primary ideals to noncommutative rings exist<ref>See the references to Chatters-Hajarnavis, Goldman, Gorton-Heatherly, and Lesieur-Croisot.</ref> but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.

==Examples and properties==
* Any [[prime ideal]] is primary, and moreover an ideal is prime if and only if it is primary and [[semiprime ideal|semiprime]].

* Every primary ideal is [[primal ideal|primal]].<ref>For the proof of the second part see the article of Fuchs</ref>

* If ''Q'' is a primary ideal, then the [[Radical of an ideal|radical]] of ''Q'' is necessarily a prime ideal ''P'', and this ideal is called the [[associated prime ideal]] of ''Q''. In this situation, ''Q'' is said to be '''''P''-primary'''.

* If ''P'' is a maximal prime ideal, then any ideal containing a power of ''P'' is ''P''-primary. Not all ''P''-primary ideals need be powers of ''P''; for example the ideal (''x'', ''y''2) is ''P''-primary for the ideal ''P'' = (''x'', ''y'') in the ring ''k''[''x'', ''y''], but is not a power of ''P''.

* In general powers of a prime ideal ''P'' need not be ''P''-primary. (An example is given by taking ''R'' to be the ring ''k''[''x'', ''y'', ''z'']/(''xy'' − ''z''2), with ''P'' the prime ideal (''x'', ''z''). If ''Q'' = ''P''2, then ''xy'' ∈ ''Q'', but ''x'' is not in ''Q'' and ''y'' is not in the radical ''P'' of ''Q'', so ''Q'' is not ''P''-primary.) However every ideal ''Q'' with radical ''P'' is contained in a smallest ''P''-primary ideal, consisting of all elements ''a'' such that ''ax'' is in ''Q'' for some ''x'' not in ''P''. In particular there is a smallest ''P''-primary ideal containing ''P''''n'', called the ''n''th '''symbolic power''' of ''P''.

* If ''A'' is a [[Noetherian ring]] and ''P'' a prime ideal, then the kernel of <math>A \to A_P</math>, the map from ''A'' to the [[localization of a ring|localization]] of ''A'' at ''P'', is the intersection of all ''P''-primary ideals.<ref>Atiyah-Macdonald, Corollary 10.21</ref>

==Footnotes==
<references/>

==References==
*{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969 |page=50}}
*{{citation |author1=Chatters, A. W. |author2=Hajarnavis, C. R. |title=Non-commutative rings with primary decomposition
|journal=Quart. J. Math. Oxford Ser. (2) |volume=22 |year=1971 |pages=73–83 |issn=0033-5606 |MR=0286822}}
*{{citation |author=Goldman, Oscar |title=Rings and modules of quotients |journal=J. Algebra |volume=13 |year=1969 |pages=10–47 |issn=0021-8693 |MR=0245608}}
*{{citation |author1=Gorton, Christine |author2=Heatherly, Henry |title=Generalized primary rings and ideals |journal=Math. Pannon. |volume=17 |year=2006 |issue=1 |pages=17–28 |issn=0865-2090 |MR=2215638}}
*[http://www.ams.org/journals/proc/1950-001-01/S0002-9939-1950-0032584-8/S0002-9939-1950-0032584-8.pdf On primal ideals], Ladislas Fuchs
*{{citation |author1=Lesieur, L. |author2=Croisot, R. |title=Algèbre noethérienne non commutative |language=French |publisher=Mémor. Sci. Math., Fasc. CLIV. Gauthier-Villars & Cie, Editeur -Imprimeur-Libraire, Paris |year=1963 |pages=119 |MR=0155861 }}

==External links==
*[http://www.encyclopediaofmath.org/index.php/Primary_ideal ''Primary ideal'' at Encyclopaedia of Mathematics]

[[Category:Commutative algebra]]
[[Category:Ideals]]

formulasearchengine - User contributions [en]

Current (mathematics)