Ideal Radical

The ideal radical of an ideal in a commutative ring with an identity element is the ideal which is the intersection of all prime ideals containing . Every proper ideal is contained in a maximal ideal, which is a prime ideal. For the unit ideal , the intersection over the empty family of prime ideals is understood to be . So the ideal radical of an ideal is always at least as big as the original ideal. Naturally, if the ideal is a prime ideal then .

Another description of the ideal radical is

This explains the connection with the radical symbol. For example, in , consider the ideal of all polynomials divisible by . Then is like a square root of . Notice that the zero set (variety) of and is the same (in because is algebraically closed). Ideal radicals are an important part of the statement of Hilbert's nullstellensatz.