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 concentrater for P.

Conversely, assume P = rad(H), and let H be a concentrater 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.

In a local ring, every ideal is M primary.

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.