primary ideal

English

Noun

primary ideal (plural primary ideals)

1. (algebra, 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.
   • 1953, D. G. Northcott, Ideal Theory, Cambridge University Press, page 10:

     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.

   • 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 v-ideals are just those in whose primary ideal decomposition only higher primary ideals occur.

   • 1997, Ralf Fröberg, An Introduction to Gröbner Bases, John Wiley & Sons, page 71:

     A primary ideal has a prime ideal as radical, so its corresponding algebraic set is irreducible. Primary ideals can, however, have multiplicity, so they give a finer description of the solution set.

Hyponyms

• (ring theory): prime ideal

Translations

(ring theory)

• German: primäres Ideal n

Further reading

• Primary decomposition on Wikipedia.Wikipedia
• Primary Ideal on Wolfram MathWorld
• Primary ideal on Encyclopedia of Mathematics