Primary Ideals, Associate Primes
Associate Primes
The associate of an ideal H, written ass(H), or σ(H),
is the set of prime ideals P1 P2 P3 etc, such that each Pi is the radical
of the conductor ideal [H:x] for some x in R.
What does this mean?
The conductor ideal is the set of elements that drive x into H.
This forms an ideal, which has a radical ideal above it.
Thus rad([H:x]) is the set of all y such that xyn lies in H.
This radical need not be prime, but if it is, it becomes an associate of H.
If x lies in H we obtain the whole ring, which cannot lead to a prime associate.
We only need consider x ∉H.
If P is minimal with respect to σ(H) it is isolated,
else it is embedded.
Intersection
Let J be the intersection of three ideals, H1 H2 and H3.
For a given x, a power of y drives x into the intersection iff it drives x into each of the three ideals.
Thus the associate of J, for a given x, is the intersection of the associates of H1 H2 and H3, for that value of x.
First Uniqueness
If J has a reduced primary decomposition H1 H2 H3 etc,
the associates of J are the corresponding prime ideals P1 P2 P3 etc.
If J lies in H, one of our P primary ideals, and x lies outside of H,
and xyn lies in H,
then a possibly higher power of y lies in H
(since H is primary),
and y lies in rad(H), which is P.
The associate defined by x lies intirely inside P.
Now if J = H, every y in P satisfies xyn ∈ H,
since yn already lies in H.
Thus P becomes the associate for [H:x].
If H is P primary, σ(H) = P.
Combine the above with intersection and assume x is not in a finite collection of primary ideals.
If J is their intersection then the associate of J, with respect to x, is the intersection of the prime ideals lying over the primary ideals.
Since the decomposition is reduced,
H1 does not contain the intersection of the remaining primary ideals.
It is possible to place x outside of H1, and inside the intersection of H2 H3 H4 etc.
The associate is now the intersection of P1 and several copies of R, which is simply P1.
each prime Pi over Hi becomes an associate of J.
If x is anywhere else, the associate is the intersection of two or more of these prime ideals.
Call this intersection Q.
If Q is not prime it does not contribute to σ(J).
If Q is prime it
must equal one of the prime ideals Pi over Hi.
Therefore σ(J) is the prime radicals of the primary ideals in the reduced decomposition of J.
You might think there are many ways to represent J as the intersection of primary ideals,
but there is a catch.
The reduced decomposition has to exhibit a specific set of prime radicals,
i.e. the associates in σ(J).
These associates depend only on J and R, and that fixes the decomposition.
J = rad(J)
Assume J = rad(J), where J is the intersection of finitely many prime ideals.
Since prime implies primary, this is a primary decomposition.
discard any duplicate primes, or primes that contain other primes.
Now if a prime contains the intersection of the others it contains their product, and contains one of them, a situation we have already dealt with.
Therefore the decomposition is reduced, and the primes become σ(J).
Next assume J = rad(J), and J has a primary decomposition.
Since radical and intersection commute, the intersection of the primes Pi yields rad(J), which is simply J.
This reproduces the situation described above.
In either case, all the primes in σ(J) are isolated, and none are embedded.
Assume J has a primary decomposition, and Q is a minimal prime containing J.
Since radical and intersection commute, rad(J) is the intersection over the primes Pi,
which lie over the primary ideals Hi.
This radical lies in Q, hence the intersection of prime ideals lies inside a prime ideal.
The intersection includes the product, hence Q contains some Pi, which contains J.
Since Q is minimal, it must equal Pi.
The minimal primes containing J are the isolated primes in σ(J).
Irreducible Components
Review the definition of spec R, as a topological space.
If J is an ideal, vJ is a closed subspace of spec R.
The irreducible components of this space correspond to the minimal primes containing J.
If J has a primary decomposition,
these are the isolated primes in σ(J).