Word.ws English Dictionary

primary ideal

primary ideal noun: given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0.

Noun

primary ideal (plural primary ideals)

  1. (ring theory) Given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0.
    • The prime and primary ideals play roles which are (very roughly) similar to those played by prime numbers and by prime.power numbers in elementary arithmetic.1953, D. G. Northcott, Ideal Theory, Cambridge University Press, page 10
    • 1970 [Frederick Ungar Publishing], John R. Schulenberger (translator), B. L. van der Waerden, Algebra, Volume 2, 1991, Springer, page 189,
      Thus all higher primary ideals are symbolic powers of higher prime ideals.
      Prüfer has called the ideals a with the property a* = a v-ideals. The integral...

Forms of primary ideal

pluralprimary ideals

Questions about primary ideal

What does primary ideal mean?
Primary ideal means given a commutative ring R, any ideal I such that for any a,b ∈ R, if ab ∈ I then either b ∈ I or aⁿ ∈ I for some integer n > 0.
What part of speech is primary ideal?
Primary ideal is a noun.
What is the plural of primary ideal?
The plural of primary ideal is primary ideals.

Words near primary ideal in the dictionary

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  2. primary forest
  3. primary goods
  4. primary healthcare
  5. primary hypothermia
  6. primary ideal
  7. primary ideals
  8. primary immunodeficiencies
  9. primary immunodeficiency
  10. primary industries
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Source and license

Adapted from Wiktionary: primary 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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