# Decomposizione primaria Decomposizione primaria (Redirected from Lasker–Noether theorem) Vai alla navigazione Vai alla ricerca In matematica, 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[Nota 1] 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.

Contenuti 1 Primary decomposition of an ideal 1.1 Esempi 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 Appunti 7 Riferimenti 8 External links Primary decomposition of an ideal Let {stile di visualizzazione R} be a Noetherian commutative ring. An ideal {stile di visualizzazione I} di {stile di visualizzazione R} is called primary if it is a proper ideal and for each pair of elements {stile di visualizzazione x} e {stile di visualizzazione y} in {stile di visualizzazione R} tale che {displaystyle xy} è dentro {stile di visualizzazione I} , o {stile di visualizzazione x} or some power of {stile di visualizzazione y} è dentro {stile di visualizzazione I} ; equivalentemente, every zero-divisor in the quotient {displaystyle R/I} is nilpotent. The radical of a primary ideal {stile di visualizzazione Q} is a prime ideal and {stile di visualizzazione Q} is said to be {stile di visualizzazione {mathfrak {p}}} -primary for {stile di visualizzazione {mathfrak {p}}={mq {Q}}} .

Permettere {stile di visualizzazione I} be an ideal in {stile di visualizzazione R} . Quindi {stile di visualizzazione I} has an irredundant primary decomposition into primary ideals: {displaystyle I=Q_{1}cap cdots cap Q_{n} } .

Irredundancy means: Removing any of the {stile di visualizzazione Q_{io}} changes the intersection, cioè. per ciascuno {stile di visualizzazione i} noi abbiamo: {displaystyle cap _{jneq i}Q_{j}not subset Q_{io}} . The prime ideals {stile di visualizzazione {mq {Q_{io}}}} are all distinct.

Inoltre, this decomposition is unique in the two ways: Il set {stile di visualizzazione {{mq {Q_{io}}}mid i}} è determinato in modo univoco da {stile di visualizzazione I} , and If {stile di visualizzazione {mathfrak {p}}={mq {Q_{io}}}} is a minimal element of the above set, poi {stile di visualizzazione Q_{io}} è determinato in modo univoco da {stile di visualizzazione I} ; infatti, {stile di visualizzazione Q_{io}} 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 {stile di visualizzazione 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 {stile di visualizzazione {{mq {Q_{io}}}mid i}} are called the prime divisors of {stile di visualizzazione I} or the primes belonging to {stile di visualizzazione I} . In the language of module theory, as discussed below, il set {stile di visualizzazione {{mq {Q_{io}}}mid i}} is also the set of associated primes of the {stile di visualizzazione R} -module {displaystyle R/I} . Esplicitamente, that means that there exist elements {stile di visualizzazione g_{1},punti ,g_{n}} in {stile di visualizzazione R} tale che {stile di visualizzazione {mq {Q_{io}}}={fin Rmid fg_{io}in I}.}  By a way of shortcut, some authors call an associated prime of {displaystyle R/I} simply an associated prime of {stile di visualizzazione I} (note this practice will conflict with the usage in the module theory).

The minimal elements of {stile di visualizzazione {{mq {Q_{io}}}mid i}} are the same as the minimal prime ideals containing {stile di visualizzazione I} and are called isolated primes. The non-minimal elements, d'altro canto, 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 {stile di visualizzazione 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 {stile di visualizzazione n} in {displaystyle mathbb {Z} } , è {displaystyle langle nrangle =langle p_{1}^{d_{1}}rangle cap cdots cap langle p_{r}^{d_{r}}sonaglio .} Allo stesso modo, in a unique factorization domain, if an element has a prime factorization {displaystyle f=up_{1}^{d_{1}}cdots p_{r}^{d_{r}},} dove {stile di visualizzazione u} is a unit, then the primary decomposition of the principal ideal generated by {stile di visualizzazione f} è {displaystyle langle frangle =langle p_{1}^{d_{1}}rangle cap cdots cap langle p_{r}^{d_{r}}sonaglio .} 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 {stile di visualizzazione k[X,y,z]} of the ideal {displaystyle I=langle x,yzrangle } è {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 {stile di visualizzazione k[X,y]} , l'ideale {angolo dello stile di visualizzazione x,si^{2}sonaglio } is a primary ideal that has {angolo dello stile di visualizzazione 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 {stile di visualizzazione k[X,y]} of the ideal {displaystyle I=langle x^{2},xyrangle } è {displaystyle I=langle x^{2},xyrangle =langle xrangle cap langle x^{2},xy,si^{n}sonaglio .} The associated primes are {displaystyle langle xrangle subset langle x,yrangle .} Esempio: 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 {stile di visualizzazione k[X,y,z],} l'ideale {displaystyle I=langle x^{2},xy,xzrangle } has the (non-unique) decomposizione primaria {displaystyle I=langle x^{2},xy,xzrangle =langle xrangle cap langle x^{2},si^{2},z^{2},xy,xz,yzrangle .} The associated prime ideals are {displaystyle langle xrangle subset langle x,y,zrangle ,} e {angolo dello stile di visualizzazione 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, tuttavia, being accessible to hand-written computation.

Permettere {stile di visualizzazione {inizio{allineato}P&=a_{0}x^{m}+un_{1}x^{m-1}y+cdots +a_{m}si^{m}\Q&=b_{0}x^{n}+b_{1}x^{n-1}y+cdots +b_{n}si^{n}fine{allineato}}} be two homogeneous polynomials in x, y, whose coefficients {stile di visualizzazione a_{1},ldot ,un_{m},b_{0},ldot ,b_{n}} are polynomials in other indeterminates {stile di visualizzazione z_{1},ldot ,z_{h}} over a field k. Questo è, P and Q belong to {displaystyle R=k[X,y,z_{1},ldot ,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 (più precisamente, it defines an algebraic set, which is a complete intersection), and thus all primary components have height two. Perciò, the associated primes of I are exactly the primes ideals of height two that contain I.

Ne consegue che {angolo dello stile di visualizzazione x,yrangle } is an associated prime of I.

Permettere {displaystyle Din k[z_{1},ldot ,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. Come {displaystyle Dnot in langle x,yrangle ,} all these monomials belong to the primary component contained in {angolo dello stile di visualizzazione x,yrangle .} This primary component contains P and Q, and the behavior of primary decompositions under localization shows that this primary component is {stile di visualizzazione {t|exists e,D^{e}tin I}.} In breve, we have a primary component, with the very simple associated prime {angolo dello stile di visualizzazione 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(io) is defined as the set of the common zeros of an ideal I of a polynomial ring {displaystyle R=k[X_{1},ldot ,X_{n}].} An irredundant primary decomposition {displaystyle I=Q_{1}cap cdots cap Q_{r}} of I defines a decomposition of V(io) into a union of algebraic sets V(Qi), which are irreducible, as not being the union of two smaller algebraic sets.

Se {stile di visualizzazione P_{io}} is the associated prime of {stile di visualizzazione Q_{io}} , poi {stile di visualizzazione V(P_{io})=V(Q_{io}),} and Lasker–Noether theorem shows that V(io) has a unique irredundant decomposition into irreducible algebraic varieties {stile di visualizzazione V(io)=bigcup V(P_{io}),} where the union is restricted to minimal associated primes. These minimal associated primes are the primary components of the radical of I. Per questa ragione, 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, in particolare, takes this approach.

Let R be a ring and M a module over it. Per definizione, an associated prime is a prime ideal appearing in the set {stile di visualizzazione {nome operatore {Anna} (m)|0neq min M}} = the set of annihilators of nonzero elements of M. Equivalentemente, a prime ideal {stile di visualizzazione {mathfrak {p}}} is an associated prime of M if there is an injection of an R-module {stile di visualizzazione 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 {nome dell'operatore dello stile di visualizzazione {Ass} _{R}(M)} o {nome dell'operatore dello stile di visualizzazione {Ass} (M)} . Directly from the definition, Se {displaystyle M=bigoplus _{io}M_{io}} , poi {nome dell'operatore dello stile di visualizzazione {Ass} (M)=bigcup _{io}nome operatore {Ass} (M_{io})} . For an exact sequence {displaystyle 0to Nto Mto Lto 0} , {nome dell'operatore dello stile di visualizzazione {Ass} (N)nome operatore sottoinsieme {Ass} (M)nome operatore sottoinsieme {Ass} (N)cup operatorname {Ass} (l)} . If R is a Noetherian ring, poi {nome dell'operatore dello stile di visualizzazione {Ass} (M)nome operatore sottoinsieme {Supp} (M)} dove {nome dell'operatore dello stile di visualizzazione {Supp} } refers to support. Anche, the set of minimal elements of {nome dell'operatore dello stile di visualizzazione {Ass} (M)} is the same as the set of minimal elements of {nome dell'operatore dello stile di visualizzazione {Supp} (M)} . 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 {stile di visualizzazione R/{mathfrak {p}}_{io}} for some prime ideals {stile di visualizzazione {mathfrak {p}}_{io}} , each of which is necessarily in the support of M. Moreover every associated prime of M occurs among the set of primes {stile di visualizzazione {mathfrak {p}}_{io}} ; cioè., {nome dell'operatore dello stile di visualizzazione {Ass} (M)sottoinsieme {{mathfrak {p}}_{1},punti ,{mathfrak {p}}_{n}}nome operatore sottoinsieme {Supp} (M)} . (In generale, these inclusions are not the equalities.) In particolare, {nome dell'operatore dello stile di visualizzazione {Ass} (M)} is a finite set when M is finitely generated.

Permettere {stile di visualizzazione M} be a finitely generated module over a Noetherian ring R and N a submodule of M. Given {nome dell'operatore dello stile di visualizzazione {Ass} (M/N)={{mathfrak {p}}_{1},punti ,{mathfrak {p}}_{n}}} , the set of associated primes of {displaystyle M/N} , there exist submodules {stile di visualizzazione Q_{io}subset M} tale che {nome dell'operatore dello stile di visualizzazione {Ass} (M/Q_{io})={{mathfrak {p}}_{io}}} e {displaystyle N=bigcap _{io=1}^{n}Q_{io}.}  A submodule N of M is called {stile di visualizzazione {mathfrak {p}}} -primary if {nome dell'operatore dello stile di visualizzazione {Ass} (M/N)={{mathfrak {p}}}} . A submodule of the R-module R is {stile di visualizzazione {mathfrak {p}}} -primary as a submodule if and only if it is a {stile di visualizzazione {mathfrak {p}}} -primary ideal; così, quando {displaystyle M=R} , the above decomposition is precisely a primary decomposition of an ideal.

Prendendo {displaystyle N=0} , the above decomposition says the set of associated primes of a finitely generated module M is the same as {stile di visualizzazione {nome operatore {Ass} (M/Q_{io})|io}} quando {displaystyle 0=cap _{1}^{n}Q_{io}} (without finite generation, there can be infinitely many associated primes.) Properties of associated primes Let {stile di visualizzazione 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). Per la stessa ragione, the union of the associated primes of an R-module M is exactly the set of zero-divisors on M, questo è, an element r such that the endomorphism {displaystyle mmapsto rm,Mto M} is not injective. Given a subset {displaystyle Phi subset operatorname {Ass} (M)} , M an R-module , there exists a submodule {displaystyle Nsubset M} tale che {nome dell'operatore dello stile di visualizzazione {Ass} (N)=nome operatore {Ass} (M)-Phi } e {nome dell'operatore dello stile di visualizzazione {Ass} (M/N)=Fi } . Permettere {displaystyle Ssubset R} be a multiplicative subset, {stile di visualizzazione M} un {stile di visualizzazione R} -module and {stile di visualizzazione Phi } the set of all prime ideals of {stile di visualizzazione R} not intersecting {stile di visualizzazione S} . Quindi {stile di visualizzazione {mathfrak {p}}mapsto S^{-1}{mathfrak {p}},,nome operatore {Ass} _{R}(M)cap Phi to operatorname {Ass} _{S^{-1}R}(S^{-1}M)} is a bijection. Anche, {nome dell'operatore dello stile di visualizzazione {Ass} _{R}(M)cap Phi =operatorname {Ass} _{R}(S^{-1}M)} . Any prime ideal minimal with respect to containing an ideal J is in {displaystyle matematica {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 matematica {Ass} (M)} consists of maximal ideals. Permettere {displaystyle Ato B} be a ring homomorphism between Noetherian rings and F a B-module that is flat over A. Quindi, for each A-module E, {nome dell'operatore dello stile di visualizzazione {Ass} _{B}(Eotimes _{UN}F)=bigcup _{{mathfrak {p}}in operatorname {Ass} (e)}nome operatore {Ass} _{B}(F/{mathfrak {p}}F)} . 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 (io : 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, è finito.

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 {displaystyle I=cap _{1}^{r}Q_{io}} (Nota: "minimo" implica {stile di visualizzazione {mq {Q_{io}}}} are distinct.) Then The set {displaystyle E=left{{mq {Q_{io}}}|1leq ileq rright}} is the set of all prime ideals in the set {stile di visualizzazione a sinistra{{mq {(io:X)}}|xin Rright}} . The set of minimal elements of E is the same as the set of minimal prime ideals over I. Inoltre, the primary ideal corresponding to a minimal prime P is the pre-image of I RP and thus is uniquely determined by I.

Adesso, 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. così, 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.

Per esempio, 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", per esempio., 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.