Primary decomposition

In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by Emanuel Lasker (1905) for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by Emmy Noether (1921).

The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The Lasker–Noether theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components.

It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. This contains the case for rings as a special case, considering the ring as a module over itself, so that ideals are submodules. This also generalizes the primary decomposition form of the structure theorem for finitely generated modules over a principal ideal domain, and for the special case of polynomial rings over a field, it generalizes the decomposition of an algebraic set into a finite union of (irreducible) varieties.

The first algorithm for computing primary decompositions for polynomial rings over a field of characteristic 0 was published by Noether's student Grete Hermann (1926). The decomposition does not hold in general for non-commutative Noetherian rings. Noether gave an example of a non-commutative Noetherian ring with a right ideal that is not an intersection of primary ideals.

Definitions

Write R for a commutative ring, and M and N for modules over it.

• A zero divisor of a module M is an element x of R such that xm = 0 for some non-zero m in M.
• An element x of R is called nilpotent in M if xⁿM = 0 for some positive integer n.
• A module M is called coprimary if every zero divisor of M is nilpotent in M. For example, groups of prime power order and free abelian groups are coprimary modules over the ring of integers.
• A submodule M of a module N is called a primary submodule if N/M is coprimary.
• An ideal I is called primary if it is a primary submodule of R. This is equivalent to saying that if ab is in I then either a is in I or bⁿ is in I for some n, and to the condition that every zero-divisor of the ring R/I is nilpotent.
• A submodule M of a module N is called irreducible if it is not an intersection of two strictly larger submodules.
• An associated prime of a module M is a prime ideal that is the annihilator of some element of M.

Statement

The Lasker–Noether theorem for modules states every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules. For the special case of ideals it states that every ideal of a Noetherian ring is a finite intersection of primary ideals.

An equivalent statement is: every finitely generated module over a Noetherian ring is contained in a finite product of coprimary modules.

The Lasker–Noether theorem follows immediately from the following three facts:

• Any submodule of a finitely generated module over a Noetherian ring is an intersection of a finite number of irreducible submodules.
• If M is an irreducible submodule of a finitely generated module N over a Noetherian ring then N/M has only one associated prime ideal.
• A finitely generated module over a Noetherian ring is coprimary if and only if it has at most one associated prime.

A proof in a somewhat different flavor is given below.

Primary decomposition of ideals

Let R be a Noetherian commutative ring, and I an ideal in R. Then I has an irredundant primary decomposition into primary ideals.

$I=Q_{1}\cap \cdots \cap Q_{n}\$

Irredundancy means:

• Removing any of the $Q_{i}$ changes the intersection, i.e.,

$Q_{1}\cap \dots \cap {\widehat {Q_{i}}}\cap \dots \cap Q_{n}\nsubseteq Q_{i}$

for all i, where the hat denotes omission.

• The associated prime ideals ${\sqrt {Q_{i}}}$ are distinct.

Moreover, this decomposition is unique in the following sense: the set of associated prime ideals is unique, and the primary ideal above every minimal prime in this set is also unique. However, primary ideals which are associated with non-minimal prime ideals are in general not unique.

In the case of the ring of integers $\mathbb {Z}$, the Lasker–Noether theorem is equivalent to the fundamental theorem of arithmetic. If an integer n has prime factorization $n=\pm p_{1}^{d_{1}}\cdots p_{r}^{d_{r}}$, then the primary decomposition of the ideal $\langle n\rangle$ generated by n in $\mathbb {Z}$, is

$\langle n\rangle =\langle p_{1}^{d_{1}}\rangle \cap \cdots \cap \langle p_{r}^{d_{r}}\rangle .$

Similarly, in a unique factorization domain, if an element has a prime factorization $f=up_{1}^{d_{1}}\cdots p_{r}^{d_{r}},$ where u is a unit, then the primary decomposition of the principal ideal generated by f is

$\langle f\rangle =\langle p_{1}^{d_{1}}\rangle \cap \cdots \cap \langle p_{r}^{d_{r}}\rangle .$

Examples

The examples of the section are designed for illustrating some properties of primary decompositions, which may appear as surprising or counter-intuitive. All examples are ideals in a polynomial ring over a field k.

Intersection vs. product

The primary decomposition in $k[x,y,z]$ of the ideal $I=\langle x,yz\rangle$ is

$I=\langle x,yz\rangle =\langle x,y\rangle \cap \langle x,z\rangle .$

Because of the generator of degree one, I is not the product of two larger ideals. A similar example is given, in two indeterminates by

$I=\langle x,y(y+1)\rangle =\langle x,y\rangle \cap \langle x,y+1\rangle .$

Primary vs. prime power

In $k[x,y]$, the ideal $\langle x,y^{2}\rangle$ is a primary ideal that has $\langle x,y\rangle$ as associated prime. It is not a power of its associated prime.

Non-uniqueness and non-minimal associated prime

For every positive integer n, a primary decomposition in $k[x,y]$ of the ideal $I=\langle x^{2},xy\rangle$ is

$I=\langle x^{2},xy\rangle =\langle x\rangle \cap \langle x^{2},xy,y^{n}\rangle .$

The associated primes are

$\langle x\rangle \subset \langle x,y\rangle .$

Non-associated prime between two associated primes

In $k[x,y,z],$ the ideal $I=\langle x^{2},xy,xz\rangle$ has the (non-unique) primary decomposition

$I=\langle x^{2},xy,xz\rangle =\langle x\rangle \cap \langle x^{2},y^{2},z^{2},xy,xz,yz\rangle .$

The associated prime ideals are $\langle x\rangle \subset \langle x,y,z\rangle ,$ and $\langle x,y\rangle$ is a non associated prime ideal such that

$\langle x\rangle \subset \langle x,y\rangle \subset \langle x,y,z\rangle .$

A complicated example

Unless for very simple examples, a primary decomposition may be hard to compute and may have a very complicated output. The following example has been designed for providing such a complicated output, and, nevertheless, being accessible to hand-written computation.

Let

${\begin{aligned}P&=a_{0}x^{m}+a_{1}x^{m-1}y+\cdots +a_{m}y^{m}\\Q&=b_{0}x^{n}+b_{1}x^{n-1}y+\cdots +b_{n}y^{n}\end{aligned}}$

be two homogeneous polynomials in x, y, whose coefficients $a_{1},\ldots ,a_{m},b_{0},\ldots ,b_{n}$ are polynomials in other indeterminates $z_{1},\ldots ,z_{h}$ over a field k. That is, P and Q belong to $R=k[x,y,z_{1},\ldots ,z_{h}],$ and this is in this ring that a primary decomposition of the ideal $I=\langle P,Q\rangle$ is searched. For computing the primary decomposition, we suppose first that 1 is a greatest common divisor of P and Q.

This condition implies that I has no primary component of height one. As I is generated by two elements, this implies that it is a complete intersection (more precisely, it defines an algebraic set, which is a complete intersection), and thus all primary components have height two. Therefore, the associated primes of I are exactly the primes ideals of height two that contain I.

It follows that $\langle x,y\rangle$ is an associated prime of I.

Let $D\in k[z_{1},\ldots ,z_{h}]$ be the homogeneous resultant in x, y of P and Q. As the greatest common divisor of P and Q is a constant, the resultant D is not zero, and resultant theory implies that I contains all products of D by a monomial in x, y of degree m + n – 1. As $D\not \in \langle x,y\rangle ,$ all these monomials belong to the primary component contained in $\langle x,y\rangle .$ This primary component contains P and Q, and the behavior of primary decompositions under localization shows that this primary component is

$\{t|\exists e,D^{e}t\in I\}.$

In short, we have a primary component, with the very simple associated prime $\langle x,y\rangle ,$ such all its generating sets involve all indeterminates.

The other primary component contains D. One may prove that if P and Q are sufficiently generic (for example if the coefficients of P and Q are distinct indeterminates), then there is only another primary component, which is a prime ideal, and is generated by P, Q and D.

Geometric interpretation

In algebraic geometry, an affine algebraic set V(I) is defined as the set of the common zeros of an ideal I of a polynomial ring $R=k[x_{1},\ldots ,x_{n}].$

An irredundant primary decomposition

$I=Q_{1}\cap \cdots \cap Q_{r}$

of I defines a decomposition of V(I) into a union of algebraic sets V(Q_i), which are irreducible, as not being the union of two smaller algebraic sets.

If $P_{i}$ is the associated prime of $Q_{i}$, then $V(P_{i})=V(Q_{i}),$ and Lasker–Noether theorem shows that V(I) has a unique irredundant decomposition into irreducible algebraic varieties

$V(I)=\bigcup V(P_{i}),$

where the union is restricted to minimal associated primes. These minimal associated primes are the primary components of the radical of I. For this reason, the primary decomposition of the radical of I is sometimes called the prime decomposition of I.

The components of a primary decomposition (as well as of the algebraic set decomposition) corresponding to minimal primes are said isolated, and the others are said embedded.

For the decomposition of algebraic varieties, only the minimal primes are interesting, but in intersection theory, and, more generally in scheme theory, the complete primary decomposition has a geometric meaning.

Proof

Nowadays, it is common to do primary decomposition within the theory of associated primes. The proof below is in the spirit of this approach.

Let M be a finitely generated module over a Noetherian ring R and N a submodule. To show N admits a primary decomposition, by replacing M by $M/N$, it is enough to show that when $N=0$. Now,

$0=\cap Q_{i}\Leftrightarrow \emptyset =\operatorname {Ass} (\cap Q_{i})=\cap \operatorname {Ass} (Q_{i})$

where $Q_{i}$ are primary submodules of M. In other words, 0 has a primary decomposition if, for each associated prime P of M, there is a primary submodule Q such that $P\not \in \operatorname {Ass} (Q)$. Now, consider the set $\{N\subseteq M|P\not \in \operatorname {Ass} (N)\}$ (which is nonempty since zero is in it). The set has a maximal element Q since M is a Noetherian module. If Q is not P-primary, say, $P'\neq P$ is associated with $M/Q$, then $R/P'\simeq Q'/Q$ for some submodule Q', contradicting the maximality. (Note: $P\not \in \operatorname {Ass} (Q)\subset \operatorname {Ass} (Q')$.) Thus, Q is primary and the proof is complete.

Remark: The same proof shows that if R, M, N are all graded, then $Q_{i}$ in the decomposition may be taken to be graded as well.

Minimal decompositions and uniqueness

In this section, all modules will be finitely generated over a Noetherian ring R.

A primary decomposition of a submodule M of a module N is called minimal if it has the smallest possible number of primary modules. For minimal decompositions, the primes of the primary modules are uniquely determined: they are the associated primes of N/M. Moreover, the primary submodules associated to the minimal or isolated associated primes (those not containing any other associated primes) are also unique. However the primary submodules associated to the non-minimal associated primes (called embedded primes for geometric reasons) need not be unique.

Example: Let N = R = k[x, y] for some field k, and let M be the ideal (xy, y²). Then M has two different minimal primary decompositions M = (y) ∩ (x, y²) = (y) ∩ (x + y, y²). The minimal prime is (y) and the embedded prime is (x, y).

Non-Noetherian case

The next theorem gives necessary and sufficient conditions for a ring to have primary decompositions for its ideals.

Theorem — Let R be a commutative ring. Then the following are equivalent.

1. Every ideal in R has a primary decomposition.
2. R has the following properties:
   • (L1) For every proper ideal I and a prime ideal P, there exists an x in R - P such that (I : x) is the pre-image of I R_P under the localization map R → R_P.
   • (L2) For every ideal I, the set of all pre-images of I S⁻¹R under the localization map R → S⁻¹R, S running over all multiplicatively closed subsets of R, is finite.

The proof is given at Chapter 4 of Atiyah–MacDonald as a series of exercises.

There is the following uniqueness theorem for an ideal having a primary decomposition.

Theorem — Let R be a commutative ring and I an ideal. Suppose I has a minimal primary decomposition $I=\cap _{1}^{r}Q_{i}$ (note: "minimal" implies ${\sqrt {Q_{i}}}$ are distinct.) Then

1. The set $E=\left\{{\sqrt {Q_{i}}}|1\leq i\leq r\right\}$ is the set of all prime ideals in the set $\left\{{\sqrt {(I:x)}}|x\in R\right\}$.
2. The set of minimal elements of E is the same as the set of minimal prime ideals over I. Moreover, the primary ideal corresponding to a minimal prime P is the pre-image of I R_P and thus is uniquely determined by I.

Now, for any commutative ring R, an ideal I and a minimal prime P over I, the pre-image of I R_P under the localization map is the smallest P-primary ideal containing I. Thus, in the setting of preceding theorem, the primary ideal Q corresponding to a minimal prime P is also the smallest P-primary ideal containing I and is called the P-primary component of I.

For example, if the power Pⁿ of a prime P has a primary decomposition, then its P-primary component is the n-th symbolic power of P.

Additive theory of ideals

This result is the first in an area now known as the additive theory of ideals, which studies the ways of representing an ideal as the intersection of a special class of ideals. The decision on the "special class", e.g., primary ideals, is a problem in itself. In the case of non-commutative rings, the class of tertiary ideals is a useful substitute for the class of primary ideals.

References

• M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN 0-201-40751-5
• Danilov, V.I. (2001) [1994], "Lasker ring", Encyclopedia of Mathematics, EMS Press
• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960, esp. section 3.3.
• Hermann, Grete (1926), "Die Frage der endlich vielen Schritte in der Theorie der Polynomideale", Mathematische Annalen, 95: 736–788, doi:10.1007/BF01206635. English translation in Communications in Computer Algebra 32/3 (1998): 8–30.
• Lasker, E. (1905), "Zur Theorie der Moduln und Ideale", Math. Ann., 60: 19–116, doi:10.1007/BF01447495
• Markov, V.T. (2001) [1994], "Primary decomposition", Encyclopedia of Mathematics, EMS Press
• Matsumura, Hideyuki (1970), Commutative algebra
• Noether, Emmy (1921), "Idealtheorie in Ringbereichen" (PDF), Mathematische Annalen, 83 (1): 24, doi:10.1007/BF01464225
• Curtis, Charles (1952), "On Additive Ideal Theory in General Rings", American Journal of Mathematics, 74 (3), The Johns Hopkins University Press: 687–700, doi:10.2307/2372273, JSTOR 2372273
• Krull, Wolfgang (1928), "Zur Theorie der zweiseitigen Ideale in nichtkommutativen Bereichen", Mathematische Zeitschrift, 28 (1): 481–503, doi:10.1007/BF01181179

External links

• http://mathoverflow.net/questions/105138/is-primary-decomposition-still-important