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In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.

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Relationship with prime elementsEdit

Irreducible elements should not be confused with prime elements. (A non-zero non-unit element   in a commutative ring   is called prime if, whenever   for some   and   in   then   or   In an integral domain, every prime element is irreducible,[1][2] but the converse is not true in general. The converse is true for unique factorization domains[2] (or, more generally, GCD domains.)

Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal. However, if   is a GCD domain, and   is an irreducible element of  , then as noted above   is prime and so the ideal generated by   is a prime ideal of  .[3]

ExampleEdit

In the quadratic integer ring   it can be shown using norm arguments that the number 3 is irreducible. However, it is not a prime element in this ring since, for example,

 

but 3 does not divide either of the two factors.[4]

See alsoEdit

ReferencesEdit

  1. ^ Consider   a prime element of   and suppose   Then   or   Say   then we have   Because   is an integral domain we have   So   is a unit and   is irreducible.
  2. ^ a b Sharpe (1987) p.54
  3. ^ "Archived copy". Archived from the original on 2010-06-20. Retrieved 2009-03-18.CS1 maint: Archived copy as title (link)
  4. ^ William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9