Word.ws English Dictionary

radical ideal

radical ideal noun: 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).

Noun

radical ideal (plural radical ideals)

  1. (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).
    • 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.1981, Harry C. Hutchins, Examples of Commutative Rings, Polygonal Publishing House, page 3
    • 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
    Synonyms:
    semiprime ideal

Synonyms of radical ideal

  • semiprime ideal

Forms of radical ideal

pluralradical ideals

Questions about radical ideal

What does radical ideal mean?
Radical ideal means 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).
What part of speech is radical ideal?
Radical ideal is a noun.
What is the plural of radical ideal?
The plural of radical ideal is radical ideals.

Words near radical ideal in the dictionary

  1. radical feminisms
  2. radical feminist
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  4. radical hysterectomies
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  6. radical ideal
  7. radical ideals
  8. radical ion
  9. radical ions
  10. radical line
  11. radical lines

Source and license

Adapted from Wiktionary: radical ideal (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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