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
- ^ 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.
- ^ ^{a} ^{b} Sharpe (1987) p.54
- ^ "Archived copy". Archived from the original on 2010-06-20. Retrieved 2009-03-18.
- ^ William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9
- Sharpe, David (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6. Zbl 0674.13008.