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 UFDs (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 the ideal generated by is a prime ideal of .
but does not divide either of the two factors.
- Consider p a prime that is reducible: p=ab. Then p | ab => p | a or p | b. Say p | a => a = pc, then we have: p=ab=pcb => p(1-cb)=0. Because R is an integral domain we have: cb=1. So b is a unit and p is irreducible.
- Sharpe (1987) p.54
- William W. Adams and Larry Joel Goldstein (1976), Introduction to Number Theory, p. 250, Prentice-Hall, Inc., ISBN 0-13-491282-9