TOPICS
Search

Prime Element


A nonzero and noninvertible element a of a ring R which generates a prime ideal. It can also be characterized by the condition that whenever a divides a product in R, a divides one of the factors. The prime elements of Z are the prime numbers P.

In an integral domain, every prime element is irreducible, but the converse holds only in unique factorization domains. The ring Z[isqrt(5)], where i is the imaginary unit, is not a unique factorization domain, and there the element 2 is irreducible, but not prime, since 2 divides the product (1-isqrt(5))(1+isqrt(5))=6, but it does not divide any of the factors.


See also

Irreducible Element, Prime Number, Unique Factorization, Unique Factorization Domain

This entry contributed by Margherita Barile

Explore with Wolfram|Alpha

Cite this as:

Barile, Margherita. "Prime Element." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PrimeElement.html

Subject classifications