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Primary ideal

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of a commutative ring $R$

An ideal $I$ of $R$ such that if $a,b \in R$ and $ab \in I$, then either $b \in I$ or $a^n \in I$ for some natural number $n$. In the ring $\mathbf{Z}$ of integers a primary ideal is an ideal of the form $p^n\mathbf{Z}$, where $p$ is a prime number and $n$ 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 $\mathrm{Ass}(M)$ denote the set of prime ideals of a ring $R$ that are annihilators of non-zero submodules of a module $M$. A submodule $N$ of a module $M$ over a Noetherian ring $R$ is called primary if $\mathrm{Ass}(M/N)$ is a one-element set. If $R$ is commutative, then every proper submodule of a Noetherian $R$-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 $N$ of a module $M$ is called primary if for every non-zero injective submodule $E_1$ of the injective hull $E$ of the module $M/N$ (cf. Injective module) the intersection of the kernels of the homomorphisms from $E$ into $E_1$ is trivial. Another successful generalization is the notion of a tertiary ideal [4]: A left ideal $I$ of a left Noetherian ring $R$ is called tertiary if, for any $a\in R$, $b \in R\setminus I$, it follows from $aRb \subseteq I$ that, for any $c \in R/I$, there is an element $d \in Rc \setminus I$ such that $aRd \subseteq I$. Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring $R$ is primary if and only if $R$ satisfies the Artin–Rees condition: For arbitrary left ideals $I,J$ of $R$ there is a natural number $n$ such that $I^n \cap J \subseteq IJ$ (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)
How to Cite This Entry:
Primary ideal. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Primary_ideal&oldid=42017
This article was adapted from an original article by V.T. Markov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article