radical ideal

English

Noun

radical ideal (plural radical ideals)

1. (algebra, commutative algebra, ring theory) An 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.

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

     It follows at once that ${\displaystyle R/I}$ is a reduced ring iff ${\displaystyle I}$ is a radical ideal. Thus, the ideals ${\displaystyle I(X)}$ are all radical ideals.

   • 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 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

• (ideal that is its own radical): semiprime ideal

Hyponyms

• (ideal that is its own radical): vanishing ideal

Translations

Further reading

• Radical of an ideal on Wikipedia.Wikipedia
• Semiprime ring on Wikipedia.Wikipedia