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Let  be a commutative ring. For any ideal  of  , the radical of  , written  or
 , is the set
 for some integer  The radical of an ideal  is always an ideal of  .
If
 , then  is called a radical ideal.
Every prime ideal is a radical ideal. If  is a radical ideal, the quotient ring  is a ring with no nonzero nilpotent elements.
More generally, the radical of an ideal in can be defined over an arbitrary ring. Let  be an ideal of a ring  , the radical of  is the set of  such that every m-system containing  has a non-empty intersection with  :
 if  is an  -system, and  then  Under this definition, we see that  is again an ideal (two-sided) and it is a subset of
 for some integer  . Furthermore, if  is commutative, the two sets coincide. In other words, this definition of a radical of an ideal is indeed a “generalization” of the radical of an ideal in a commutative ring.
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