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To define it, consider first that in any commutative ring R the nilpotent elements form an ideal N: this can be checked directly from the definitions using the binomial theorem. We call N the nilpotent radical or nilradical of R. If r is an element of R and I is an ideal in R, we can call r nilpotent mod I if r maps to the nilpotent radical of the factor ring R/I. The set of all these elements r then automatically forms an ideal in R, which we call the radical of I. Put more simply, the radical of I consists of the r in R, some power of which lies in I.
We see that the nilradical of R is nothing but the radical of the zero ideal {0}.
It can be shown, as an application of Zorn's lemma, that the radical of I is the intersection of all the prime ideals of R that contain I.
The intersection of all maximal ideals in R is the Jacobson radical of R; it always contains the nilradical of R. If the ring R is finitely generated as a ring over Z, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I is equal to the intersection of all the maximal ideals of R that contain I.