Radical ideal

English

Noun

1. Template:Lb An Template:L I within a ring R that is its own Template:L (i.e., for any r ∈ R, if rⁿ ∈ I for some Template:L Template:L 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.

   • Template:Quote-book
   • 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 idealsTemplate:....

Synonyms

• Template:Sense Template:L

Hyponyms

• Template:Sense Template:L

Further reading

• Template:Pedia
• Template:Pedia