The radical of an ideal a in a ring R is the ideal which is the intersection of all prime ideals containing a. 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 a is prime then r(a) = a. Another description of the radical r(a) is r(a) = {x:x^n element a for some integer n>0}.
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