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A representation of an [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744501.png" /> of a [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744502.png" /> (or of a submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744503.png" /> of a [[Module|module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744504.png" />) as an intersection of primary ideals (primary submodules, cf. [[Primary ideal|Primary ideal]]). The primary decomposition generalizes the factorization of an integer into a product of powers of distinct prime numbers. The existence of primary decompositions in a polynomial ring was proved by E. Lasker [[#References|[1]]], and in an arbitrary commutative [[Noetherian ring|Noetherian ring]] by E. Noether [[#References|[2]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744505.png" /> be a commutative Noetherian ring. A primary decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744506.png" /> is called irreducible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744507.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744508.png" /> and if the radicals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p0744509.png" /> of the ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445010.png" /> are pairwise distinct (the radical of a primary ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445011.png" /> is the unique [[Prime ideal|prime ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445013.png" /> for some natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445014.png" />). The set of prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445015.png" /> is uniquely determined by the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445016.png" /> (the first uniqueness theorem for primary decompositions). The minimal elements (with respect to inclusion) of this set are called the isolated prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445017.png" />, the other elements are called the imbedded prime ideals. The primary ideals corresponding to isolated prime ideals are also uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445018.png" /> (the second uniqueness theorem for primary decompositions, cf. [[#References|[3]]]). The isolated prime ideals of an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445019.png" /> of a polynomial ring over a field correspond to the irreducible components of the [[Affine variety|affine variety]] of roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074450/p07445020.png" />. There are various generalizations of the notion of primary decomposition. The axiomatization of primary decompositions led to the development of the [[Additive theory of ideals|additive theory of ideals]].
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A representation of an [[ideal]] $I$ of a [[ring]] $R$ (or of a submodule $N$ of a [[module]] $M$) as an intersection of primary ideals (primary submodules, cf. [[Primary ideal]]). The primary decomposition generalizes the factorization of an integer into a product of powers of distinct prime numbers. The existence of primary decompositions in a polynomial ring was proved by E. Lasker [[#References|[1]]], and in an arbitrary commutative [[Noetherian ring]] by E. Noether [[#References|[2]]]. Let $R$ be a commutative Noetherian ring. A primary decomposition $I = \cap_{i=1}^n Q_i$ is called irreducible if $\cap_{i\ne j}Q_i \ne I$ for any $j = 1,\ldots,n$ and if the radicals $P_1,\ldots,P_n$ of the ideals $Q_1,\ldots,Q_n$ are pairwise distinct (the radical of a primary ideal $Q$ is the unique [[prime ideal]] $P$ such that $P^n \subseteq Q$ for some natural number $n$). The set of prime ideals $\{P_1,\ldots,P_n\}$ is uniquely determined by the ideal $I$ (the first uniqueness theorem for primary decompositions). The minimal elements (with respect to inclusion) of this set are called the isolated prime ideals of $I$, the other elements are called the imbedded prime ideals. The primary ideals corresponding to isolated prime ideals are also uniquely determined by $I$ (the second uniqueness theorem for primary decompositions, cf. [[#References|[3]]]). The isolated prime ideals of an ideal $I$ of a polynomial ring over a field correspond to the irreducible components of the [[affine variety]] of roots of $I$. There are various generalizations of the notion of primary decomposition. The axiomatization of primary decompositions led to the development of the [[additive theory of ideals]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lasker,  "Zur Theorie der Moduln und Ideale"  ''Math. Ann.'' , '''60'''  (1905)  pp. 20–116</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Noether,  "Idealtheorie in Ringbereichen"  ''Math. Ann.'' , '''83'''  (1921)  pp. 24–66</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.F. Atiyah,  I.G. Macdonald,  "Introduction to commutative algebra" , Addison-Wesley  (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1–2''' , Springer  (1975)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Lasker,  "Zur Theorie der Moduln und Ideale"  ''Math. Ann.'' , '''60'''  (1905)  pp. 20–116</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  E. Noether,  "Idealtheorie in Ringbereichen"  ''Math. Ann.'' , '''83'''  (1921)  pp. 24–66</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  M.F. Atiyah,  I.G. Macdonald,  "Introduction to commutative algebra" , Addison-Wesley  (1969)</TD></TR>
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<TR><TD valign="top">[4]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1–2''' , Springer  (1975)</TD></TR>
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<TR><TD valign="top">[5]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR>
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</table>
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Latest revision as of 20:06, 5 October 2017

A representation of an ideal $I$ of a ring $R$ (or of a submodule $N$ of a module $M$) as an intersection of primary ideals (primary submodules, cf. Primary ideal). The primary decomposition generalizes the factorization of an integer into a product of powers of distinct prime numbers. The existence of primary decompositions in a polynomial ring was proved by E. Lasker [1], and in an arbitrary commutative Noetherian ring by E. Noether [2]. Let $R$ be a commutative Noetherian ring. A primary decomposition $I = \cap_{i=1}^n Q_i$ is called irreducible if $\cap_{i\ne j}Q_i \ne I$ for any $j = 1,\ldots,n$ and if the radicals $P_1,\ldots,P_n$ of the ideals $Q_1,\ldots,Q_n$ are pairwise distinct (the radical of a primary ideal $Q$ is the unique prime ideal $P$ such that $P^n \subseteq Q$ for some natural number $n$). The set of prime ideals $\{P_1,\ldots,P_n\}$ is uniquely determined by the ideal $I$ (the first uniqueness theorem for primary decompositions). The minimal elements (with respect to inclusion) of this set are called the isolated prime ideals of $I$, the other elements are called the imbedded prime ideals. The primary ideals corresponding to isolated prime ideals are also uniquely determined by $I$ (the second uniqueness theorem for primary decompositions, cf. [3]). The isolated prime ideals of an ideal $I$ of a polynomial ring over a field correspond to the irreducible components of the affine variety of roots of $I$. There are various generalizations of the notion of primary decomposition. The axiomatization of primary decompositions led to the development of the additive theory of ideals.

References

[1] E. Lasker, "Zur Theorie der Moduln und Ideale" Math. Ann. , 60 (1905) pp. 20–116
[2] E. Noether, "Idealtheorie in Ringbereichen" Math. Ann. , 83 (1921) pp. 24–66
[3] M.F. Atiyah, I.G. Macdonald, "Introduction to commutative algebra" , Addison-Wesley (1969)
[4] O. Zariski, P. Samuel, "Commutative algebra" , 1–2 , Springer (1975)
[5] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
How to Cite This Entry:
Primary decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primary_decomposition&oldid=14571
This article was adapted from an original article by V.T. Markov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article