Primary Ideals, Saturating with a Single Element

Saturating with a Single Element

Let J have a primary decomposition, and designate a finite set of primes P₁ P₂ P₃ etc in σ(J). This set must be closed under containment. If it includes P₃, and P₃ contains P₇, then our set must also include P₇.

Assume there is an element f that is in all the other primes Q₁ Q₂ Q₃ etc of σ(J). In other words, f is in each Q_i, and not in any P_i. The intersection of the primary ideals under P_i equals the saturation of J with respect to f.

This is practically a corollary of the previous theorem, which completely characterizes the saturations of J. The powers of f cannot lie in one of our primes P_i, else f lies in P_i. Thus the multiplicatively closed set seeded by f misses our designated primes, and collides with the other primes. Saturation by f produces the intersection of our designated primary ideals. But we would like to establish a specific power of f, so there is a little more work to do.

Let x lie in each H_i below P_i. Consider an off prime Q_i. Since f is in Q_i, which is the radical of its H_i, some power of f winds up in H_i. When this is multiplied by x, the result lies in H_i.

Switch to a prime P_i in our set. Now x already lies in H_i, so x times f⁰ lies in H_i. Put this all together and a sufficiently high power of f, say fⁿ, takes x into each primary ideal, and into J.

Notice that n does not depend on x. Every x in the intersection of the designated primary ideals is driven into J by fⁿ.

Intersection agrees with saturation in the fraction ring, whose denominators are powers of f. Expressed another way, the intersection is the conductor ideal [J:fⁿ].

Such an f must exist. Suppose it does not. Let W be the intersection of the prime ideals Q_i, the "other" primes. Clearly W is in each Q_i. If any part of W misses all the primes in P_i, we have found our element f. Thus W is contained in the finite union of prime ideals P_i, and W lies in exactly one of them.

Let W lie in P₃. Thus the intersection of the primes Q_i lies in P₃. This means some prime, say Q₇, lies in P₃. However, all the primes contained in P₃ are part of our designated set. We have reached a contradiction, hence there is an f that meets our criteria. We can use this f to saturate J, and obtain the intersection of our primary ideals.

As a special case, suppose all the primes in σ(J) are in our designated set. There are no "other" primes, and f is not well defined. However, we don't have to saturate J at all, because J is already the intersection of the designated primary ideals. Set f = 1, and the theorem is satisfied.