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Radical of an ideal

In ring theory (a branch of 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 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.