radical ideal

Noun

 * 1)  An  I within a ring R that is its own  (i.e., for any r ∈ R, if rn ∈ I for some   n, then r ∈ I).
 * 2) * 1981, Harry C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, page 3,
 * Any intersection of radical ideals is again a radical ideal. Since not every intersection of prime ideals is a prime ideal, it follows that not all radical ideals are prime.
 * 1) * 1997, D. D. Anderson, Dong Je Kwak, Some Remarks on G-Noetherian Rings, Paul-Jean Cahen, Marco Fontana, Evan Houston, Salah-Eddine Kabbaj (editors), Commutative Ring Theory: Proceedings of the II International Conference, Marcel Dekker, page 30,
 * Thus Theorem 2 [8] is a special case of the well-known result that a commutative ring R satisfies the ascending chain condition on radical ideals if and only if R satisfies the ascending chain condition on prime ideals and each radical ideal of R is a finite intersection of prime ideals.
 * Thus Theorem 2 [8] is a special case of the well-known result that a commutative ring R satisfies the ascending chain condition on radical ideals if and only if R satisfies the ascending chain condition on prime ideals and each radical ideal of R is a finite intersection of prime ideals.