Prime Element (original) (raw)

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

In an integral domain, every prime element is irreducible, but the converse holds only in unique factorization domains. The ring , 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, 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

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

Subject classifications

• Algebra
• Ring Theory
• MathWorld Contributors
• Barile