In ring (mathematics) various concepts of radical are introduced. For the purposes of commutative algebra one notion is usually sufficient, the idea of the radical of an ideal I in a commutative ring R.

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 of R. For any ideal I we can call r nilpotent mod I if r maps to the nilpotent radical of R/I - this then automatically decribes an ideal 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.

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.