Editing Irreducible element (section)

== Relationship with prime elements ==
Irreducible elements should not be confused with [[prime element]]s. (A non-zero non-unit element <math>a</math> in a [[commutative ring]] <math>R</math> is called prime if, whenever <math>a \mid bc</math> for some <math>b</math> and <math>c</math> in <math>R,</math> then <math>a \mid b</math> or <math>a \mid c.</math>) In an [[integral domain]], every prime element is irreducible,{{efn|Consider <math>p</math> a prime element of <math>R</math> and suppose <math>p=ab.</math> Then <math>p \mid ab,</math> so <math> p \mid a</math> or <math>p \mid b.</math> Say <math>p \mid a,</math> so <math> a = pc</math> for some <math> c \in R </math>. Then we have <math>p=ab=pcb,</math> and so <math>p(1-cb)=0.</math> Because <math>R</math> is an integral domain we have <math>cb=1.</math> Therefore <math>b</math> is a unit and <math>p</math> is irreducible.}}<ref name=Sha54>{{cite book | last=Sharpe | first=David | title=Rings and factorization | url=https://archive.org/details/ringsfactorizati0000shar | url-access=registration | zbl=0674.13008 | publisher=[[Cambridge University Press]] | year=1987 | isbn=0-521-33718-6 | page = 54 }}</ref> but the converse is not true in general. The converse is true for [[unique factorization domain]]s<ref name=Sha54/> (or, more generally, [[GCD domain]]s).

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 <math>D</math> is a GCD domain and <math>x</math> is an irreducible element of <math>D</math>, then as noted above <math>x</math> is prime, and so the ideal generated by <math>x</math> is a prime (hence irreducible) ideal of <math>D</math>.