radical of an ideal

If an ideal is a set of 'rules for being zero', its radical is the same rule made forgetful of multiplicity. The ideal (x^2) demands that x vanish to second order; its radical (x) demands only that x vanish at all. Taking the radical strips away the infinitesimal thickening and keeps just the underlying set of conditions — exactly the information geometry can see.

Given an ideal I of a commutative ring R, its radical sqrt(I) is the set of all elements a such that a^n lies in I for some positive integer n. This is again an ideal, it contains I, and taking the radical twice changes nothing: sqrt(sqrt(I)) = sqrt(I). An ideal equal to its own radical is called a radical ideal. The radical of I can also be described intrinsically as the intersection of all prime ideals containing I.

The radical of the zero ideal is the nilradical. Prime ideals are automatically radical, but radical ideals need not be prime — (xy) in k[x, y] is radical (it equals sqrt((xy))) yet not prime, since it is the intersection of the two primes (x) and (y). Under the Nullstellensatz over an algebraically closed field, radical ideals are precisely those that arise as the full ideal of some algebraic set.