radical ideal

plural radical ideals

1. commutative algebra; ring theory; commutative algebra; ring theoryalgebraAn ideal I within a ring R that is its own radical (i.e., for any r ∈ R, if rⁿ ∈ I for some positive integer n, then r ∈ I).

   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.

   2 more examples

   It follows at once that R#47;I is a reduced ring iff I is a radical ideal. Thus, the ideals I(X) are all radical ideals.

   1995, David Eisenbud, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, page 33

   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 […] .

   Synonyms: semiprime ideal