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 for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by .
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[1] was published by Noether's student .[2] [3] 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.
Let
R
I
R
x
y
R
xy
I
x
y
I
R/I
Q
Q
ak{p}
ak{p}=\sqrt{Q}
Let
I
R
I
I=Q1\cap … \capQn
Irredundancy means:
Qi
i
\capjQj\not\subsetQi
\sqrt{Qi}
Moreover, this decomposition is unique in the two ways:
\{\sqrt{Qi}\midi\}
I
ak{p}=\sqrt{Qi}
Qi
I
Qi
IRak{p
R\toRak{p
I
The elements of
\{\sqrt{Qi}\midi\}
I
I
\{\sqrt{Qi}\midi\}
R
R/I
g1,...,gn
R
\sqrt{Qi}=\{f\inR\midfgi\inI\}.
By a way of shortcut, some authors call an associated prime of
R/I
I
\{\sqrt{Qi}\midi\}
I
In the case of the ring of integers
Z
n
n=\pm
d1 | |
p | |
1 |
…
dr | |
p | |
r |
\langlen\rangle
n
Z
\langlen\rangle=\langle
d1 | |
p | |
1 |
\rangle\cap … \cap\langle
dr | |
p | |
r |
\rangle.
Similarly, in a unique factorization domain, if an element has a prime factorization
f=u
d1 | |
p | |
1 |
…
dr | |
p | |
r |
,
u
f
\langlef\rangle=\langle
d1 | |
p | |
1 |
\rangle\cap … \cap\langle
dr | |
p | |
r |
\rangle.
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 .
The primary decomposition in
k[x,y,z]
I=\langlex,yz\rangle
I=\langlex,yz\rangle=\langlex,y\rangle\cap\langlex,z\rangle.
Because of the generator of degree one, is not the product of two larger ideals. A similar example is given, in two indeterminates by
I=\langlex,y(y+1)\rangle=\langlex,y\rangle\cap\langlex,y+1\rangle.
In
k[x,y]
\langlex,y2\rangle
\langlex,y\rangle
For every positive integer, a primary decomposition in
k[x,y]
I=\langlex2,xy\rangle
I=\langlex2,xy\rangle=\langlex\rangle\cap\langlex2,xy,yn\rangle.
The associated primes are
\langlex\rangle\subset\langlex,y\rangle.
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 decompositionsM = (y) ∩ (x, y2) = (y) ∩ (x + y, y2).The minimal prime is (y) and the embedded prime is (x, y).
In
k[x,y,z],
I=\langlex2,xy,xz\rangle
I=\langlex2,xy,xz\rangle=\langlex\rangle\cap\langlex2,y2,z2,xy,xz,yz\rangle.
\langlex\rangle\subset\langlex,y,z\rangle,
\langlex,y\rangle
\langlex\rangle\subset\langlex,y\rangle\subset\langlex,y,z\rangle.
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
m | |
{align} P&=a | |
0x |
+
m-1 | |
a | |
1x |
y+ …
m | |
+a | |
my |
n | |
\\ Q&=b | |
0x |
+
n-1 | |
b | |
1x |
y+ …
n \end | |
+b | |
ny |
{align}
a1,\ldots,am,b0,\ldots,bn
z1,\ldots,zh
R=k[x,y,z1,\ldots,zh],
I=\langleP,Q\rangle
This condition implies that has no primary component of height one. As 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 are exactly the primes ideals of height two that contain .
It follows that
\langlex,y\rangle
Let
D\ink[z1,\ldots,zh]
D\not\in\langlex,y\rangle,
\langlex,y\rangle.
\{t|\existse,Det\inI\}.
In short, we have a primary component, with the very simple associated prime
\langlex,y\rangle,
The other primary component contains . One may prove that if and are sufficiently generic (for example if the coefficients of and are distinct indeterminates), then there is only another primary component, which is a prime ideal, and is generated by, and .
R=k[x1,\ldots,xn].
An irredundant primary decomposition
I=Q1\cap … \capQr
If
Pi
Qi
V(Pi)=V(Qi),
V(I)=cupV(Pi),
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 .
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.
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 which is the annihilator of a nonzero element of M; that is,
ak{p}=\operatorname{Ann}(m)
m\inM
m\ne0
ak{p}
R/ak{p}\hookrightarrowM
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, there exists an associated prime of M if and only if M is nonzero.
The set of associated primes of M is denoted by
\operatorname{Ass}R(M)
\operatorname{Ass}(M)
M=oplusiMi
\operatorname{Ass}(M)=cupi\operatorname{Ass}(Mi)
0\toN\toM\toL\to0
\operatorname{Ass}(N)\subset\operatorname{Ass}(M)\subset\operatorname{Ass}(N)\cup\operatorname{Ass}(L)
\operatorname{Ass}(M)\subset\operatorname{Supp}(M)
\operatorname{Supp}
\operatorname{Ass}(M)
\operatorname{Supp}(M)
If M is a finitely generated module over R, then there is a finite ascending sequence of submodules
0=M0\subsetneqM1\subsetneq … \subsetneqMn-1\subsetneqMn=M
R/ak{p}i
ak{p}i
ak{p}i
\operatorname{Ass}(M)\subset\{ak{p}1,...,ak{p}n\}\subset\operatorname{Supp}(M)
\operatorname{Ass}(M)
Let
M
\operatorname{Ass}(M/N)=\{ak{p}1,...,ak{p}n\}
M/N
Qi\subsetM
\operatorname{Ass}(M/Qi)=\{ak{p}i\}
N=
n | |
cap | |
i=1 |
Qi.
\sqrt{Qi}=(I:gi)
M/N
N=0
0=\capQi\iff\emptyset=\operatorname{Ass}(\capQi)=\cap\operatorname{Ass}(Qi)
Qi
P\not\in\operatorname{Ass}(Q)
\{N\subseteqM|P\not\in\operatorname{Ass}(N)\}
P'\neP
M/Q
R/P'\simeqQ'/Q
Remark: The same proof shows that if R, M, N are all graded, then
Qi
ak{p}
\operatorname{Ass}(M/N)=\{ak{p}\}
ak{p}
ak{p}
M=R
Taking
N=0
\{\operatorname{Ass}(M/Qi)|i\}
0=
n | |
\cap | |
1 |
Qi
Let
R
m\mapstorm,M\toM
\Phi\subset\operatorname{Ass}(M)
N\subsetM
\operatorname{Ass}(N)=\operatorname{Ass}(M)-\Phi
\operatorname{Ass}(M/N)=\Phi
S\subsetR
M
R
\Phi
R
S
\operatorname{Ass}R(M)\cap\Phi=
-1 | |
\operatorname{Ass} | |
R(S |
M)
AssR(R/J).
Ass(M)
A\toB
\operatorname{Ass}B(E ⊗ AF)=cupak{p\in\operatorname{Ass}(E)}\operatorname{Ass}B(F/ak{p}F)
The next theorem gives necessary and sufficient conditions for a ring to have primary decompositions for its ideals.
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.
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. 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.
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.
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