Irreducible element

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In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.

Relationship with prime elements[edit]

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 (hence irreducible) ideal of .[3]


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[edit]


  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.{{cite web}}: 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