# Primary ideal

A **primary ideal** is a proper ideal \( \mathfrak{q} \) of a commutative ring \( R \) such that whenever a product \( ab \) lies in \( \mathfrak{q} \), either \( a \in \mathfrak{q} \) or some power \( b^{n} \) with \( n \ge 1 \) lies in \( \mathfrak{q} \).<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> The notion, introduced by [Emanuel Lasker](https://www.edgepedia.org/emanuel-lasker) in 1905 in the setting of polynomial rings, replaces powers of prime numbers in the ring-theoretic analog of integer factorization: in a ring, powers of prime ideals are not the right building blocks, primary ideals are.<sup>[2](https://mathshistory.st-andrews.ac.uk/HistTopics/Ring_theory/)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( ab \in \mathfrak{q} \) implies \( a \in \mathfrak{q} \) or \( b^{n} \in \mathfrak{q} \) for some \( n \ge 1 \); equivalently \( b \in \sqrt{\mathfrak{q}} \)<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> |
| Quotient characterization | \( \mathfrak{q} \) is primary if and only if every zero divisor in \( R/\mathfrak{q} \) is nilpotent<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> |
| Radical | The radical of a primary ideal is a prime ideal, called the associated prime; \( \mathfrak{q} \) is then said to be \( \mathfrak{p} \)-primary<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> |
| Integers | In \( \mathbb{Z} \) the primary ideals are exactly \( (p^{n}) = p^{n}\mathbb{Z} \) for prime numbers \( p \)<sup>[3](https://www.math.clemson.edu/~macaule/classes/f23_math8510/slides/math8510_slides_chapter08_h.pdf)</sup> |
| Existence theorem | Every proper ideal of a commutative Noetherian ring is a finite intersection of primary ideals (Lasker 1905 for polynomial rings; Emmy Noether 1921 in full generality)<sup>[4](https://encyclopediaofmath.org/wiki/Primary_decomposition)</sup><sup> • </sup><sup>[5](https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/)</sup>  |
| Uniqueness | The set of associated primes is uniquely determined by the ideal; the primary components belonging to isolated (minimal) primes are also unique<sup>[4](https://encyclopediaofmath.org/wiki/Primary_decomposition)</sup> |
| Computational status | Even for monomial ideals in a polynomial ring over a field, finding a primary decomposition is NP-complete<sup>[6](https://eloisagrifo.github.io/SymbolicPowers.pdf)</sup> |

## Definition and equivalent forms

The defining condition can be restated in the language of radicals: \( \mathfrak{q} \) is primary if whenever \( ab \in \mathfrak{q} \), either \( a \in \mathfrak{q} \) or \( b \in \sqrt{\mathfrak{q}} \), where \( \sqrt{\mathfrak{q}} \) denotes the radical of \( \mathfrak{q} \), the set of elements some power of which lies in \( \mathfrak{q} \).<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> Every prime ideal is primary, since in that case \( \sqrt{\mathfrak{q}} = \mathfrak{q} \).<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup>

A second characterization passes to the quotient ring: a proper ideal \( \mathfrak{q} \) is primary if and only if every [zero divisor](https://www.edgepedia.org/zero-divisor) in \( R/\mathfrak{q} \) is nilpotent.<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> This explains the name by analogy with prime numbers. In the integers, the condition that a product be divisible by \( p^{n} \) while one factor is not forces the other factor to be divisible by \( p \); the primary ideals of \( \mathbb{Z} \) are precisely the ideals \( (p^{n}) \).<sup>[3](https://www.math.clemson.edu/~macaule/classes/f23_math8510/slides/math8510_slides_chapter08_h.pdf)</sup>

## The associated prime

The radical of any primary ideal is a prime ideal.<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> The proof is short: if \( ab \in \sqrt{\mathfrak{q}} \), then \( a^{k}b^{k} \in \mathfrak{q} \) for some \( k \), so primarity gives \( a^{k} \in \mathfrak{q} \) or \( b^{kn} \in \mathfrak{q} \), and in either case one of \( a, b \) lies in \( \sqrt{\mathfrak{q}} \).<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> This prime \( \mathfrak{p} = \sqrt{\mathfrak{q}} \) is called the **associated prime** of \( \mathfrak{q} \), and \( \mathfrak{q} \) is said to be \( \mathfrak{p} \)-primary; it is the smallest prime ideal of \( R \) containing \( \mathfrak{q} \).<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup>

The converse fails: a primary ideal need not be a power of its associated prime, and a power of a prime ideal need not be primary. In the ring \( R = F[x,y,z]/(xy - z^{2}) \) over a field \( F \), the ideal \( \mathfrak{p} = (x,z) \) is prime, but \( \mathfrak{p}^{2} \) is not primary: \( xy \in \mathfrak{p}^{2} \) while \( x \notin \mathfrak{p}^{2} \) and \( y \notin \sqrt{\mathfrak{p}^{2}} = \mathfrak{p} \).<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> Two reliable cases do hold: if \( \sqrt{I} \) is maximal then \( I \) is primary, so any power of a maximal ideal is primary with that maximal ideal as associated prime.<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> And a finite intersection of primary ideals with the same associated prime is again primary.<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup>

A standard example of a primary ideal that is not prime is \( (x^{2}, y) \) in \( F[x,y] \): the quotient is isomorphic to \( F[x]/(x^{2}) \), in which every zero divisor is nilpotent, and the associated prime is \( (x,y) \).<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup>

## Primary decomposition

The importance of primary ideals comes from the **Lasker–Noether theorem**: every proper ideal of a commutative [Noetherian ring](https://www.edgepedia.org/noetherian-ring) can be written as a finite intersection of primary ideals.<sup>[4](https://encyclopediaofmath.org/wiki/Primary_decomposition)</sup><sup> • </sup><sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup>  Lasker proved the existence of such decompositions for polynomial rings in 1905, and [Emmy Noether](https://www.edgepedia.org/emmy-noether) extended the result to arbitrary commutative rings with the ascending chain condition on ideals in her 1921 paper *Idealtheorie in Ringbereichen*.<sup>[4](https://encyclopediaofmath.org/wiki/Primary_decomposition)</sup><sup> • </sup><sup>[5](https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/)</sup>  The decomposition generalizes factorization of an integer into a product of powers of distinct primes.<sup>[4](https://encyclopediaofmath.org/wiki/Primary_decomposition)</sup>

A decomposition is called **minimal** (or irredundant) if the radicals of the primary components are pairwise distinct and no component can be dropped; every decomposable ideal has a minimal decomposition, obtained by removing redundant components and merging those with the same associated prime.<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> Uniqueness is partial. The set of associated primes \( \{\mathfrak{p}_{1}, \ldots, \mathfrak{p}_{n}\} \) is uniquely determined by the ideal \( I \) (the first uniqueness theorem), and the primary components corresponding to the minimal elements of this set, the **isolated primes**, are uniquely determined as well (the second uniqueness theorem).<sup>[4](https://encyclopediaofmath.org/wiki/Primary_decomposition)</sup> The remaining associated primes are called **embedded**, and their primary components are not unique: for example, the ideal \( (xy, y^{2}) \) of \( F[x,y] \) has both \( (x,y)^{2} \cap (y) \) and \( (x, y^{2}) \cap (y) \) as minimal decompositions.<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup> The non-uniqueness of primary decompositions led B. L. van der Waerden in 1931 to introduce an equivalence relation on ideals coarser than equality.<sup>[7](https://encyclopediaofmath.org/wiki/Commutative_algebra)</sup>

One route to the existence theorem runs through irreducible ideals, those that cannot be written as an intersection of two strictly larger ideals. In a Noetherian ring every irreducible ideal is primary, and every proper ideal is a finite intersection of irreducible ideals; combining the two statements gives the primary decomposition theorem.<sup>[1](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)</sup>

## Geometry

For an ideal \( I \) of a polynomial ring over a field, the isolated primes of \( I \) correspond to the irreducible components of the affine variety defined by \( I \).<sup>[4](https://encyclopediaofmath.org/wiki/Primary_decomposition)</sup> In this geometric reading, a primary ideal describes a variety together with specified infinitesimal thickening, and the decomposition of \( I \) splits the solution set of the polynomial equations into irreducible pieces with multiplicities.

## Computation

Computing a primary decomposition from generators of an ideal is a fundamental problem of computational algebra, and a difficult one: even restricted to monomial ideals in a polynomial ring over a field, finding a primary decomposition is NP-complete.<sup>[6](https://eloisagrifo.github.io/SymbolicPowers.pdf)</sup> Algorithms in general use include the method of Shimoyama and Yokoyama and the algorithm of Eisenbud, Huneke, and Vasconcelos, together with a hybrid of the two, and special algorithms for monomial and binomial ideals.<sup>[6](https://eloisagrifo.github.io/SymbolicPowers.pdf)</sup> The computer algebra system [Macaulay2](https://www.edgepedia.org/macaulay2) implements these strategies in its `primaryDecomposition` function, which returns an irredundant decomposition whose components have radicals equal to the associated primes of the input ideal.<sup>[8](https://macaulay2.com/doc/Macaulay2-1.25.11/share/doc/Macaulay2/PrimaryDecomposition/html/toc.html)</sup><sup> • </sup><sup>[9](https://sources.debian.org/src/macaulay2/1.26.06%2Bds-3/M2/Macaulay2/packages/PrimaryDecomposition/doc.m2)</sup>  Extensions of the theory continue to be developed, including \( S \)-primary decompositions in \( S \)-Noetherian rings and decompositions of ideals depending on parameters.<sup>[10](https://arxiv.org/abs/2401.00922)</sup><sup> • </sup><sup>[11](https://arxiv.org/abs/2408.15917)</sup>

## References

1. [Romyar Sharifi, Abstract Algebra, Chapter 11: Commutative algebra (UCLA lecture notes)](https://math.ucla.edu/~sharifi/notes/algebra-ch11.html)
2. [Ring Theory, MacTutor History of Mathematics, University of St Andrews](https://mathshistory.st-andrews.ac.uk/HistTopics/Ring_theory/)
3. [Macauley, Chapter 8: Rings, Math 8510 Visual Algebra, Clemson University](https://www.math.clemson.edu/~macaule/classes/f23_math8510/slides/math8510_slides_chapter08_h.pdf)
4. [Primary decomposition, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Primary_decomposition)
5. [Emmy Noether (1882–1935), Biography, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/)
6. [Grifo et al., Symbolic powers lecture notes (primary decomposition in Macaulay2; NP-completeness of monomial case)](https://eloisagrifo.github.io/SymbolicPowers.pdf)
7. [Commutative algebra, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Commutative_algebra)
8. [PrimaryDecomposition package, Macaulay2 documentation](https://macaulay2.com/doc/Macaulay2-1.25.11/share/doc/Macaulay2/PrimaryDecomposition/html/toc.html)
9. [Macaulay2 PrimaryDecomposition package source, Debian Sources](https://sources.debian.org/src/macaulay2/1.26.06%2Bds-3/M2/Macaulay2/packages/PrimaryDecomposition/doc.m2)
10. [S-Primary Decomposition in S-Noetherian Rings, arXiv:2401.00922](https://arxiv.org/abs/2401.00922)
11. [Comprehensive Systems for Primary Decompositions of Parametric Ideals, arXiv:2408.15917](https://arxiv.org/abs/2408.15917)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Ideals and factorizations of ideals*

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