# Primary decomposition

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 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[Note 1] was published by Noether's student Grete Hermann (1926).[1][2] 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.

Contents 1 Primary decomposition of an ideal 1.1 Examples 1.1.1 Intersection vs. product 1.1.2 Primary vs. prime power 1.1.3 Non-uniqueness and embedded prime 1.1.4 Non-associated prime between two associated primes 1.1.5 A complicated example 1.2 Geometric interpretation 2 Primary decomposition from associated primes 3 Properties of associated primes 4 Non-Noetherian case 5 Additive theory of ideals 6 Notes 7 References 8 External links Primary decomposition of an ideal Let {displaystyle R} be a Noetherian commutative ring. An ideal {displaystyle I} of {displaystyle R} is called primary if it is a proper ideal and for each pair of elements {displaystyle x} and {displaystyle y} in {displaystyle R} such that {displaystyle xy} is in {displaystyle I} , either {displaystyle x} or some power of {displaystyle y} is in {displaystyle I} ; equivalently, every zero-divisor in the quotient {displaystyle R/I} is nilpotent. The radical of a primary ideal {displaystyle Q} is a prime ideal and {displaystyle Q} is said to be {displaystyle {mathfrak {p}}} -primary for {displaystyle {mathfrak {p}}={sqrt {Q}}} .

Let {displaystyle I} be an ideal in {displaystyle R} . Then {displaystyle I} has an irredundant primary decomposition into primary ideals: {displaystyle I=Q_{1}cap cdots cap Q_{n} } .

Irredundancy means: Removing any of the {displaystyle Q_{i}} changes the intersection, i.e. for each {displaystyle i} we have: {displaystyle cap _{jneq i}Q_{j}not subset Q_{i}} . The prime ideals {displaystyle {sqrt {Q_{i}}}} are all distinct.

Moreover, this decomposition is unique in the two ways: The set {displaystyle {{sqrt {Q_{i}}}mid i}} is uniquely determined by {displaystyle I} , and If {displaystyle {mathfrak {p}}={sqrt {Q_{i}}}} is a minimal element of the above set, then {displaystyle Q_{i}} is uniquely determined by {displaystyle I} ; in fact, {displaystyle Q_{i}} is the pre-image of {displaystyle IR_{mathfrak {p}}} under the localization map {displaystyle Rto R_{mathfrak {p}}} .

Primary ideals which correspond to non-minimal prime ideals over {displaystyle I} are in general not unique (see an example below). For the existence of the decomposition, see #Primary decomposition from associated primes below.

The elements of {displaystyle {{sqrt {Q_{i}}}mid i}} are called the prime divisors of {displaystyle I} or the primes belonging to {displaystyle I} . In the language of module theory, as discussed below, the set {displaystyle {{sqrt {Q_{i}}}mid i}} is also the set of associated primes of the {displaystyle R} -module {displaystyle R/I} . Explicitly, that means that there exist elements {displaystyle g_{1},dots ,g_{n}} in {displaystyle R} such that {displaystyle {sqrt {Q_{i}}}={fin Rmid fg_{i}in I}.} [3] By a way of shortcut, some authors call an associated prime of {displaystyle R/I} simply an associated prime of {displaystyle I} (note this practice will conflict with the usage in the module theory).

The minimal elements of {displaystyle {{sqrt {Q_{i}}}mid i}} are the same as the minimal prime ideals containing {displaystyle I} and are called isolated primes. The non-minimal elements, on the other hand, are called the embedded primes.

In the case of the ring of integers {displaystyle mathbb {Z} } , the Lasker–Noether theorem is equivalent to the fundamental theorem of arithmetic. If an integer {displaystyle n} has prime factorization {displaystyle n=pm p_{1}^{d_{1}}cdots p_{r}^{d_{r}}} , then the primary decomposition of the ideal {displaystyle langle nrangle } generated by {displaystyle n} in {displaystyle mathbb {Z} } , is {displaystyle langle nrangle =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 {displaystyle f=up_{1}^{d_{1}}cdots p_{r}^{d_{r}},} where {displaystyle u} is a unit, then the primary decomposition of the principal ideal generated by {displaystyle f} is {displaystyle langle frangle =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 {displaystyle k[x,y,z]} of the ideal {displaystyle I=langle x,yzrangle } is {displaystyle I=langle x,yzrangle =langle x,yrangle cap langle x,zrangle .} 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 {displaystyle I=langle x,y(y+1)rangle =langle x,yrangle cap langle x,y+1rangle .} Primary vs. prime power In {displaystyle k[x,y]} , the ideal {displaystyle langle x,y^{2}rangle } is a primary ideal that has {displaystyle langle x,yrangle } as associated prime. It is not a power of its associated prime.

Non-uniqueness and embedded prime For every positive integer n, a primary decomposition in {displaystyle k[x,y]} of the ideal {displaystyle I=langle x^{2},xyrangle } is {displaystyle I=langle x^{2},xyrangle =langle xrangle cap langle x^{2},xy,y^{n}rangle .} The associated primes are {displaystyle langle xrangle subset langle x,yrangle .} 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-associated prime between two associated primes In {displaystyle k[x,y,z],} the ideal {displaystyle I=langle x^{2},xy,xzrangle } has the (non-unique) primary decomposition {displaystyle I=langle x^{2},xy,xzrangle =langle xrangle cap langle x^{2},y^{2},z^{2},xy,xz,yzrangle .} The associated prime ideals are {displaystyle langle xrangle subset langle x,y,zrangle ,} and {displaystyle langle x,yrangle } is a non associated prime ideal such that {displaystyle langle xrangle subset langle x,yrangle subset langle x,y,zrangle .} 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 {displaystyle {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 {displaystyle a_{1},ldots ,a_{m},b_{0},ldots ,b_{n}} are polynomials in other indeterminates {displaystyle z_{1},ldots ,z_{h}} over a field k. That is, P and Q belong to {displaystyle R=k[x,y,z_{1},ldots ,z_{h}],} and it is in this ring that a primary decomposition of the ideal {displaystyle I=langle P,Qrangle } 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 {displaystyle langle x,yrangle } is an associated prime of I.

Let {displaystyle Din 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 {displaystyle Dnot in langle x,yrangle ,} all these monomials belong to the primary component contained in {displaystyle langle x,yrangle .} This primary component contains P and Q, and the behavior of primary decompositions under localization shows that this primary component is {displaystyle {t|exists e,D^{e}tin I}.} In short, we have a primary component, with the very simple associated prime {displaystyle langle x,yrangle ,} 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 {displaystyle R=k[x_{1},ldots ,x_{n}].} An irredundant primary decomposition {displaystyle I=Q_{1}cap cdots cap Q_{r}} of I defines a decomposition of V(I) into a union of algebraic sets V(Qi), which are irreducible, as not being the union of two smaller algebraic sets.

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

Primary decomposition from associated primes Nowadays, it is common to do primary decomposition of ideals and modules within the theory of associated primes. Bourbaki's influential textbook Algèbre commutative, in particular, takes this approach.

Let R be a ring and M a module over it. By definition, an associated prime is a prime ideal appearing in the set {displaystyle {operatorname {Ann} (m)|0neq min M}} = the set of annihilators of nonzero elements of M. Equivalently, a prime ideal {displaystyle {mathfrak {p}}} is an associated prime of M if there is an injection of an R-module {displaystyle R/{mathfrak {p}}hookrightarrow M} .

A maximal element of the set of annihilators of nonzero elements of M can be shown to be a prime ideal and thus, when R is a Noetherian ring, M is nonzero if and only if there exists an associated prime of M.

The set of associated primes of M is denoted by {displaystyle operatorname {Ass} _{R}(M)} or {displaystyle operatorname {Ass} (M)} . Directly from the definition, If {displaystyle M=bigoplus _{i}M_{i}} , then {displaystyle operatorname {Ass} (M)=bigcup _{i}operatorname {Ass} (M_{i})} . For an exact sequence {displaystyle 0to Nto Mto Lto 0} , {displaystyle operatorname {Ass} (N)subset operatorname {Ass} (M)subset operatorname {Ass} (N)cup operatorname {Ass} (L)} .[4] If R is a Noetherian ring, then {displaystyle operatorname {Ass} (M)subset operatorname {Supp} (M)} where {displaystyle operatorname {Supp} } refers to support.[5] Also, the set of minimal elements of {displaystyle operatorname {Ass} (M)} is the same as the set of minimal elements of {displaystyle operatorname {Supp} (M)} .[5] If M is a finitely generated module over R, then there is a finite ascending sequence of submodules {displaystyle 0=M_{0}subsetneq M_{1}subsetneq cdots subsetneq M_{n-1}subsetneq M_{n}=M,} such that each quotient Mi /Mi−1 is isomorphic to {displaystyle R/{mathfrak {p}}_{i}} for some prime ideals {displaystyle {mathfrak {p}}_{i}} , each of which is necessarily in the support of M.[6] Moreover every associated prime of M occurs among the set of primes {displaystyle {mathfrak {p}}_{i}} ; i.e., {displaystyle operatorname {Ass} (M)subset {{mathfrak {p}}_{1},dots ,{mathfrak {p}}_{n}}subset operatorname {Supp} (M)} .[7] (In general, these inclusions are not the equalities.) In particular, {displaystyle operatorname {Ass} (M)} is a finite set when M is finitely generated.

Let {displaystyle M} be a finitely generated module over a Noetherian ring R and N a submodule of M. Given {displaystyle operatorname {Ass} (M/N)={{mathfrak {p}}_{1},dots ,{mathfrak {p}}_{n}}} , the set of associated primes of {displaystyle M/N} , there exist submodules {displaystyle Q_{i}subset M} such that {displaystyle operatorname {Ass} (M/Q_{i})={{mathfrak {p}}_{i}}} and {displaystyle N=bigcap _{i=1}^{n}Q_{i}.} [8][9] A submodule N of M is called {displaystyle {mathfrak {p}}} -primary if {displaystyle operatorname {Ass} (M/N)={{mathfrak {p}}}} . A submodule of the R-module R is {displaystyle {mathfrak {p}}} -primary as a submodule if and only if it is a {displaystyle {mathfrak {p}}} -primary ideal; thus, when {displaystyle M=R} , the above decomposition is precisely a primary decomposition of an ideal.

Taking {displaystyle N=0} , the above decomposition says the set of associated primes of a finitely generated module M is the same as {displaystyle {operatorname {Ass} (M/Q_{i})|i}} when {displaystyle 0=cap _{1}^{n}Q_{i}} (without finite generation, there can be infinitely many associated primes.) Properties of associated primes Let {displaystyle R} be a Noetherian ring. Then The set of zero-divisors on R is the same as the union of the associated primes of R (this is because the set of zerodivisors of R is the union of the set of annihilators of nonzero elements, the maximal elements of which are associated primes).[10] For the same reason, the union of the associated primes of an R-module M is exactly the set of zero-divisors on M, that is, an element r such that the endomorphism {displaystyle mmapsto rm,Mto M} is not injective.[11] Given a subset {displaystyle Phi subset operatorname {Ass} (M)} , M an R-module , there exists a submodule {displaystyle Nsubset M} such that {displaystyle operatorname {Ass} (N)=operatorname {Ass} (M)-Phi } and {displaystyle operatorname {Ass} (M/N)=Phi } .[12] Let {displaystyle Ssubset R} be a multiplicative subset, {displaystyle M} an {displaystyle R} -module and {displaystyle Phi } the set of all prime ideals of {displaystyle R} not intersecting {displaystyle S} . Then {displaystyle {mathfrak {p}}mapsto S^{-1}{mathfrak {p}},,operatorname {Ass} _{R}(M)cap Phi to operatorname {Ass} _{S^{-1}R}(S^{-1}M)} is a bijection.[13] Also, {displaystyle operatorname {Ass} _{R}(M)cap Phi =operatorname {Ass} _{R}(S^{-1}M)} .[14] Any prime ideal minimal with respect to containing an ideal J is in {displaystyle mathrm {Ass} _{R}(R/J).} These primes are precisely the isolated primes. A module M over R has finite length if and only if M is finitely generated and {displaystyle mathrm {Ass} (M)} consists of maximal ideals.[15] Let {displaystyle Ato B} be a ring homomorphism between Noetherian rings and F a B-module that is flat over A. Then, for each A-module E, {displaystyle operatorname {Ass} _{B}(Eotimes _{A}F)=bigcup _{{mathfrak {p}}in operatorname {Ass} (E)}operatorname {Ass} _{B}(F/{mathfrak {p}}F)} .[16] 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.

Every ideal in R has a primary decomposition. 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 RP under the localization map R → RP. (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.[17] 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 {displaystyle I=cap _{1}^{r}Q_{i}} (note: "minimal" implies {displaystyle {sqrt {Q_{i}}}} are distinct.) Then The set {displaystyle E=left{{sqrt {Q_{i}}}|1leq ileq rright}} is the set of all prime ideals in the set {displaystyle left{{sqrt {(I:x)}}|xin Rright}} . 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 RP 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 RP under the localization map is the smallest P-primary ideal containing I.[18] 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.

Notes ^ Primary decomposition requires testing irreducibility of polynomials, which is not always algorithmically possible in nonzero characteristic. ^ Ciliberto, Ciro; Hirzebruch, Friedrich; Miranda, Rick; Teicher, Mina, eds. (2001). Applications of Algebraic Geometry to Coding Theory, Physics and Computation. Dordrecht: Springer Netherlands. ISBN 978-94-010-1011-5. ^ Hermann, G. (1926). "Die Frage der endlich vielen Schritte in der Theorie der Polynomideale". Mathematische Annalen (in German). 95: 736–788. doi:10.1007/BF01206635. ^ In other words, {displaystyle {sqrt {Q_{i}}}=(I:g_{i})} is the ideal quotient. ^ Bourbaki, Ch. IV, § 1, no 1, Proposition 3. ^ Jump up to: a b Bourbaki, Ch. IV, § 1, no 3, Corollaire 1. ^ Bourbaki, Ch. IV, § 1, no 4, Théorème 1. ^ Bourbaki, Ch. IV, § 1, no 4, Théorème 2. ^ Bourbaki, Ch. IV, § 2, no. 2. Theorem 1. ^ Here is the proof of the existence of the decomposition (following Bourbaki). 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 {displaystyle M/N} , it is enough to show that when {displaystyle N=0} . Now, {displaystyle 0=cap Q_{i}iff emptyset =operatorname {Ass} (cap Q_{i})=cap operatorname {Ass} (Q_{i})} where {displaystyle 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 {displaystyle Pnot in operatorname {Ass} (Q)} . Now, consider the set {displaystyle {Nsubseteq M|Pnot 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, {displaystyle P'neq P} is associated with {displaystyle M/Q} , then {displaystyle R/P'simeq Q'/Q} for some submodule Q', contradicting the maximality. Thus, Q is primary and the proof is complete. Remark: The same proof shows that if R, M, N are all graded, then {displaystyle Q_{i}} in the decomposition may be taken to be graded as well. ^ Bourbaki, Ch. IV, § 1, Corollary 3. ^ Bourbaki, Ch. IV, § 1, Corollary 2. ^ Bourbaki, Ch. IV, § 1, Proposition 4. ^ Bourbaki, Ch. IV, § 1, no. 2, Proposition 5. ^ Matsumura 1970, 7.C Lemma ^ Cohn, P. M. (2003), Basic Algebra, Springer, Exercise 10.9.7, p. 391, ISBN 9780857294289. ^ Bourbaki, Ch. IV, § 2. Theorem 2. ^ Atiyah–MacDonald 1969 ^ Atiyah–MacDonald 1969, Ch. 4. Exercise 11 References M. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison–Wesley, 1994. ISBN 0-201-40751-5 Bourbaki, Algèbre commutative. 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 (in German), 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", Mathematische Annalen, 83 (1): 24–66, doi:10.1007/BF01464225 Curtis, Charles (1952), "On Additive Ideal Theory in General Rings", American Journal of Mathematics, The Johns Hopkins University Press, 74 (3): 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 "Is primary decomposition still important?". MathOverflow. August 21, 2012. Categories: Commutative algebraTheorems in ring theoryAlgebraic geometry

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