Ideal Radical

The radical of an ideal in a ring is the ideal which is the intersection of all prime ideals containing . Note that any ideal is contained in a maximal ideal, which is always prime. So the radical of an ideal is always at least as big as the original ideal. Naturally, if the ideal is prime then .

Another description of the radical is

This explains the connection with the radical symbol. For example, in , consider the ideal of all polynomials with degree at least 2. Then is like a square root of . Notice that the zero set (variety) of and is the same (in because is algebraically closed). Radicals are an important part of the statement of Hilbert's Nullstellensatz.