# Primary decomposition

A representation of an ideal of a ring (or of a submodule of a module ) 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 be a commutative Noetherian ring. A primary decomposition is called irreducible if for any and if the radicals of the ideals are pairwise distinct (the radical of a primary ideal is the unique prime ideal such that for some natural number ). The set of prime ideals is uniquely determined by the ideal (the first uniqueness theorem for primary decompositions). The minimal elements (with respect to inclusion) of this set are called the isolated prime ideals of , the other elements are called the imbedded prime ideals. The primary ideals corresponding to isolated prime ideals are also uniquely determined by (the second uniqueness theorem for primary decompositions, cf. [3]). The isolated prime ideals of an ideal of a polynomial ring over a field correspond to the irreducible components of the affine variety of roots of . 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) |

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Primary decomposition.

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