Primary Ideals, The Radical of a Primary Ideal

The Radical of a Primary Ideal

Recall that the radical of an ideal is the intersection of prime ideals containing said ideal. Thus the radical of a prime ideal P is P, as that is the intersection of all prime ideals containing P. In this section we will show that the radical of a primary ideal is prime.

Let xy lie in rad(H), while x does not lie in rad(H). This means (xy)n lies in H, but no power of x lies in H. Seen another way, xnyn lies in H, but xn does not, hence some power of yn lies in H, a power of y lies in H, and y lies in rad(H). Therefore rad(H) is a prime ideal.

Though H has its own prime radical, a prime ideal may be the radical of several different primary ideals. For example the primary ideals generated by Pn in Z all have the same prime radical P.

Assume P = rad(H) and H is primary. Now if xy is in H and x is not in H then yn is in H, and y is in P. We normally say xy in P means at least one factor is in P, but now, if xy is in H at least one factor is in P. Think of H as a concentrator for P.

Conversely, assume P = rad(H), and let H be a concentrator for P. If x is not in H then y is in P is in rad(H), yn lies in H, and H is primary.

When H is primary and P = rad(H) we say H is P primary. Here is a simple theorem about M primary ideals, where M is maximal.

Powers of M

Assume rad(H) = M for M maximal in R. In the quotient ring R/H, the image of M is a nil ideal. Let x be an element in R-M, thus x represents an element in R/H that is not in the image of M. By maximality, x and M span 1. Write cx+y = 1, where c is in R and y is in M. In the quotient ring, cx = 1-y, where y is nilpotent. Yet 1-y is a unit, hence x is a unit. All elements are nilpotent or units, R/H is a primary ring, and H is a primary ideal.

Since M = rad(H), H is M primary.

As a corollary, all powers of M are M primary. We proved this for a pid, but it holds in any ring.

Intersecting P Primary Ideals

Consider a finite set of P primary ideals H1 H2 H3 etc, and let J be their intersection. Since P contains J, rad(J) = P, or something smaller. If some other prime ideal Q contains J it contains the product of the primary ideals, and it contains at least one of them, hence it comtains P after all. Therefore rad(J) = P.

Let J contain xy, but not x. Thus there is some primary ideal Hi that contains xy, but not x. Since Hi contains a power of y, P contains y. Since rad(J) = P, J contains a power of y, and J is P primary.

Examples

A primary ideal need not be a prime power, and a prime power need not be primary.

Let R = Z[x], and let 2 and x generate P, while 4 and x generate Q. Since R/P = Z2, P is maximal. Since R/Q = Z4, Q is primary. The image of P in R/Q gives the nil radical, so P = rad(Q), and Q is P primary. Yet P2, generated by 4, 2x, and x2, lies properly inside Q, hence Q is not a power of P.

Let R = K[x,y,z] mod z2-xy, and let x and z generate P. The quotient R/P is K[y], an integral domain, so P is indeed prime. The product xy lies in P2, being equal to z2. However, x is not in P2, and no power of y lies in P2, hence P2 is not primary.