primary ideal

plural primary ideals

1. ring theoryalgebraGiven 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

   2 more examples

   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.

   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.

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