Radicals, Radical Ideal

Radical Ideal

Let H be an ideal of R. As a set, let rad(H) be the elements x, such that every m-system containing x intersects H.

If a prime P contains H and misses x, x is in the m-system R-P, and is not in rad(H), hence rad(H) lies in every P containing H. Conversely, if x is in every prime containing H, i.e. the intersection of the primes containing H, try to embed x in an m-system S missing H. Drive H up to a maximal ideal missing S, which is prime, and x is not in the intersection after all. Therefore rad(H) is the intersection of the prime ideals containing H, and rad(H) is an ideal. This definition agrees with the definition of rad(H) in a commutative ring.

Maximal Ideals are Prime

If H is proper, it misses the m-system 1, and can be raised to a maximal, prime ideal in R containing H. Thus rad(H) is always a proper ideal.

Let H be any maximal ideal, and raise H to a maximal ideal missing 1. Thus every maximal ideal is prime.

A minimal ideal need not be prime. Let P² generate H inside Z_P³.

Radical Ideal iff Semiprime

Let H be semiprime. We know H lies in rad(H), so suppose x is in rad(H)-H. Now x is in an n-system missing H, and x seeds an m-system lying entirely within the n-system. Thus x lies outside a prime containing H, and x does not lie in rad(H) after all. We have H = rad(H), and a semiprime ideal is a radical ideal.

Conversely, let H = rad(H), the intersection of all prime ideals containing H. Since a prime ideal is semiprime, H is the intersection of semiprime ideals, and H is semiprime.

Characterizing rad(H)

Let H be any proper ideal of R, and let J be the intersection of semiprime ideals containing H, hence J is semiprime. Since prime ideals are semiprime, rad(H) contains J contains H. As shown above, J = rad(J), so rad(H) cannot be larger than J. Therefore rad(H) = J.

In summary, rad(H) is the least semiprime ideal containing H, or the intersection of semiprime ideals containing H, or the intersection of prime ideals containing H.