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. For any ideal I we can call r nilpotent mod I if r maps to the nilpotent radical of the factor ring R/I - this then automatically describes an ideal in R 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.
The radical of the ring R is the nilradical.
It can be shown, as an application of Zorn's lemma, that the radical of I also is the intersection of all the maximal ideals of R that contain I.