Irreducible element

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.

Relationship with prime elements

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]

Example

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 also

References

  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. 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
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