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*.