# Decomposição primária

Decomposição primária (Redirected from Lasker–Noether theorem) Ir para a navegação Ir para a pesquisa Em matemática, 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 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[Observação 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.

Conteúdo 1 Primary decomposition of an ideal 1.1 Exemplos 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 Notas 7 Referências 8 External links Primary decomposition of an ideal Let {estilo de exibição R} be a Noetherian commutative ring. An ideal {estilo de exibição I} do {estilo de exibição R} is called primary if it is a proper ideal and for each pair of elements {estilo de exibição x} e {estilo de exibição y} dentro {estilo de exibição R} de tal modo que {displaystyle xy} é em {estilo de exibição I} , qualquer {estilo de exibição x} or some power of {estilo de exibição y} é em {estilo de exibição I} ; equivalentemente, every zero-divisor in the quotient {displaystyle R/I} is nilpotent. The radical of a primary ideal {estilo de exibição Q} is a prime ideal and {estilo de exibição Q} is said to be {estilo de exibição {mathfrak {p}}} -primary for {estilo de exibição {mathfrak {p}}={quadrado {Q}}} .

Deixar {estilo de exibição I} be an ideal in {estilo de exibição R} . Então {estilo de exibição 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_{eu}} changes the intersection, ou seja. para cada {estilo de exibição eu} temos: {displaystyle cap _{jneq i}Q_{j}not subset Q_{eu}} . The prime ideals {estilo de exibição {quadrado {Q_{eu}}}} are all distinct.

Além disso, this decomposition is unique in the two ways: The set {estilo de exibição {{quadrado {Q_{eu}}}mid i}} é determinado exclusivamente por {estilo de exibição I} , and If {estilo de exibição {mathfrak {p}}={quadrado {Q_{eu}}}} is a minimal element of the above set, então {displaystyle Q_{eu}} é determinado exclusivamente por {estilo de exibição I} ; na verdade, {displaystyle Q_{eu}} 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 {estilo de exibição 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 {estilo de exibição {{quadrado {Q_{eu}}}mid i}} are called the prime divisors of {estilo de exibição I} or the primes belonging to {estilo de exibição I} . In the language of module theory, as discussed below, the set {estilo de exibição {{quadrado {Q_{eu}}}mid i}} is also the set of associated primes of the {estilo de exibição R} -module {displaystyle R/I} . Explicitamente, that means that there exist elements {estilo de exibição g_{1},pontos ,g_{n}} dentro {estilo de exibição R} de tal modo que {estilo de exibição {quadrado {Q_{eu}}}={fin Rmid fg_{eu}in I}.} [3] By a way of shortcut, some authors call an associated prime of {displaystyle R/I} simply an associated prime of {estilo de exibição I} (note this practice will conflict with the usage in the module theory).

The minimal elements of {estilo de exibição {{quadrado {Q_{eu}}}mid i}} are the same as the minimal prime ideals containing {estilo de exibição I} and are called isolated primes. The non-minimal elements, por outro lado, are called the embedded primes.

In the case of the ring of integers {estilo de exibição mathbb {Z} } , the Lasker–Noether theorem is equivalent to the fundamental theorem of arithmetic. If an integer {estilo de exibição m} 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 {estilo de exibição m} dentro {estilo de exibição mathbb {Z} } , é {displaystyle langle nrangle =langle p_{1}^{d_{1}}rangle cap cdots cap langle p_{r}^{d_{r}}chocalho .} De forma similar, in a unique factorization domain, if an element has a prime factorization {displaystyle f=up_{1}^{d_{1}}cdots p_{r}^{d_{r}},} Onde {estilo de exibição você} is a unit, then the primary decomposition of the principal ideal generated by {estilo de exibição f} é {displaystyle langle frangle =langle p_{1}^{d_{1}}rangle cap cdots cap langle p_{r}^{d_{r}}chocalho .} 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 {estilo de exibição k[x,y,z]} of the ideal {displaystyle I=langle x,yzrangle } é {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 {estilo de exibição k[x,y]} , o ideal {estilo de exibição lang x,^{2}chocalho } is a primary ideal that has {estilo de exibição lang 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 {estilo de exibição k[x,y]} of the ideal {displaystyle I=langle x^{2},xyrangle } é {displaystyle I=langle x^{2},xyrangle =langle xrangle cap langle x^{2},xy,^{n}chocalho .} The associated primes are {displaystyle langle xrangle subset langle x,yrangle .} Exemplo: Let N = R = k[x, y] for some field k, and let M be the ideal (xy, ano 2). Then M has two different minimal primary decompositions M = (y) (x, ano 2) = (y) (x + y, ano 2). The minimal prime is (y) and the embedded prime is (x, y).

Non-associated prime between two associated primes In {estilo de exibição k[x,y,z],} o ideal {displaystyle I=langle x^{2},xy,xzrangle } has the (non-unique) decomposição primária {displaystyle I=langle x^{2},xy,xzrangle =langle xrangle cap langle x^{2},^{2},z^{2},xy,xz,yzrangle .} The associated prime ideals are {displaystyle langle xrangle subset langle x,y,zrangle ,} e {estilo de exibição lang 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, e, mesmo assim, being accessible to hand-written computation.

Deixar {estilo de exibição {começar{alinhado}P&=a_{0}x^{m}+uma_{1}x^{m-1}y+cdots +a_{m}^{m}\Q&=b_{0}x^{n}+b_{1}x^{n-1}y+cdots +b_{n}^{n}fim{alinhado}}} be two homogeneous polynomials in x, y, whose coefficients {estilo de exibição a_{1},ldots ,uma_{m},b_{0},ldots ,b_{n}} are polynomials in other indeterminates {estilo de exibição z_{1},ldots ,z_{h}} over a field k. Aquilo é, 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 (mais precisamente, it defines an algebraic set, which is a complete intersection), and thus all primary components have height two. Portanto, the associated primes of I are exactly the primes ideals of height two that contain I.

Segue que {estilo de exibição lang x,yrangle } is an associated prime of I.

Deixar {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. Como {displaystyle Dnot in langle x,yrangle ,} all these monomials belong to the primary component contained in {estilo de exibição lang x,yrangle .} This primary component contains P and Q, and the behavior of primary decompositions under localization shows that this primary component is {estilo de exibição {t|exists e,D^{e}tin I}.} In short, we have a primary component, with the very simple associated prime {estilo de exibição lang 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(EU) 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(EU) into a union of algebraic sets V(Qi), which are irreducible, as not being the union of two smaller algebraic sets.

Se {estilo de exibição P_{eu}} is the associated prime of {displaystyle Q_{eu}} , então {estilo de exibição V(P_{eu})=V(Q_{eu}),} and Lasker–Noether theorem shows that V(EU) has a unique irredundant decomposition into irreducible algebraic varieties {estilo de exibição V(EU)=bigcup V(P_{eu}),} where the union is restricted to minimal associated primes. These minimal associated primes are the primary components of the radical of I. Por esta razão, 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, e, 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, em particular, takes this approach.

Let R be a ring and M a module over it. Por definição, an associated prime is a prime ideal appearing in the set {estilo de exibição {nome do operador {Ana} (m)|0neq min M}} = the set of annihilators of nonzero elements of M. Equivalentemente, a prime ideal {estilo de exibição {mathfrak {p}}} is an associated prime of M if there is an injection of an R-module {estilo de exibição 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.

Deixar {estilo de exibição M} be a finitely generated module over a Noetherian ring R and N a submodule of M. Given {nome do operador de estilo de exibição {Ass} (M/N)={{mathfrak {p}}_{1},pontos ,{mathfrak {p}}_{n}}} , the set of associated primes of {displaystyle M/N} , there exist submodules {displaystyle Q_{eu}subset M} de tal modo que {nome do operador de estilo de exibição {Ass} (M/Q_{eu})={{mathfrak {p}}_{eu}}} e {displaystyle N=bigcap _{i=1}^{n}Q_{eu}.} [8][9] A submodule N of M is called {estilo de exibição {mathfrak {p}}} -primary if {nome do operador de estilo de exibição {Ass} (M/N)={{mathfrak {p}}}} . A submodule of the R-module R is {estilo de exibição {mathfrak {p}}} -primary as a submodule if and only if it is a {estilo de exibição {mathfrak {p}}} -primary ideal; portanto, quando {displaystyle M=R} , the above decomposition is precisely a primary decomposition of an ideal.

Tirando {displaystyle N=0} , the above decomposition says the set of associated primes of a finitely generated module M is the same as {estilo de exibição {nome do operador {Ass} (M/Q_{eu})|eu}} quando {displaystyle 0=cap _{1}^{n}Q_{eu}} (without finite generation, there can be infinitely many associated primes.) Properties of associated primes Let {estilo de exibição 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] Pela mesma razão, the union of the associated primes of an R-module M is exactly the set of zero-divisors on M, isso é, 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} de tal modo que {nome do operador de estilo de exibição {Ass} (N)=nome do operador {Ass} (M)-Phi } e {nome do operador de estilo de exibição {Ass} (M/N)=Phi } .[12] Deixar {displaystyle Ssubset R} be a multiplicative subset, {estilo de exibição M} um {estilo de exibição R} -module and {estilo de exibição Phi } the set of all prime ideals of {estilo de exibição R} not intersecting {estilo de exibição S} . Então {estilo de exibição {mathfrak {p}}mapsto S^{-1}{mathfrak {p}},,nome do operador {Ass} _{R}(M)cap Phi to operatorname {Ass} _{S^{-1}R}(S^{-1}M)} is a bijection.[13] Também, {nome do operador de estilo de exibição {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 {matemática de estilo de exibição {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 {matemática de estilo de exibição {Ass} (M)} consists of maximal ideals.[15] Deixar {displaystyle Ato B} be a ring homomorphism between Noetherian rings and F a B-module that is flat over A. Então, for each A-module E, {nome do operador de estilo de exibição {Ass} _{B}(Eotimes _{UMA}F)=bigcup _{{mathfrak {p}}in operatorname {Ass} (E)}nome do operador {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 (EU : 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, é finito.

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_{eu}} (Nota: "minimal" implica {estilo de exibição {quadrado {Q_{eu}}}} are distinct.) Then The set {displaystyle E=left{{quadrado {Q_{eu}}}|1leq ileq rright}} is the set of all prime ideals in the set {estilo de exibição à esquerda{{quadrado {(EU:x)}}|xin Rright}} . The set of minimal elements of E is the same as the set of minimal prime ideals over I. Além disso, the primary ideal corresponding to a minimal prime P is the pre-image of I RP and thus is uniquely determined by I.

Agora, 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] Desta forma, 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.

Por exemplo, 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", por exemplo., 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.