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, or equivalently, if every factoring of such element contains at least one unit.
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, but the converse is not true in general. The converse is true for unique factorization domains (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 .
but 3 does not divide either of the two factors.
- 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.
- Sharpe (1987) p.54
- "Archived copy". Archived from the original on 2010-06-20. Retrieved 2009-03-18.CS1 maint: archived copy as title (link)
- William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9