# Primary ideal

*of a commutative ring *

An ideal such that if and , then either or for some natural number . In the ring of integers a primary ideal is an ideal of the form , where is a prime number and is a natural number. In commutative algebra an important role is played by the representation of an arbitrary ideal of a commutative Noetherian ring as an intersection of a finite number of primary ideals — a primary decomposition. More generally, let denote the set of prime ideals of a ring that are annihilators of non-zero submodules of a module . A submodule of a module over a Noetherian ring is called primary if is a one-element set. If is commutative, then every proper submodule of a Noetherian -module that cannot be represented as an intersection of submodules strictly containing it is primary. In the non-commutative case this is not true and therefore attempts have been undertaken to construct various non-commutative generalizations of the notion of primarity. E.g., a proper submodule of a module is called primary if for every non-zero injective submodule of the injective hull of the module (cf. Injective module) the intersection of the kernels of the homomorphisms from into is trivial. Another successful generalization is the notion of a tertiary ideal [4]: A left ideal of a left Noetherian ring is called tertiary if, for any , , it follows from that, for any , there is an element such that . Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring is primary if and only if satisfies the Artin–Rees condition: For arbitrary left ideals of there is a natural number such that (cf. [3]).

#### References

[1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |

[2] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |

[3] | O. Goldman, "Rings and modules of quotients" J. of Algebra , 13 : 1 (1969) pp. 10–47 |

[4] | L. Lesieur, R. Croisot, "Algèbre noethérienne noncommutative" , Gauthier-Villars (1963) |

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Primary ideal.

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