Show that a is irreducible if and only if the principal ideal (a) is maximal in the set {(b) | b a nonzero nonunit in R}.

Question

Let R be a domain and $a$ a nonzero nonunit in R. Show that $a$ is irreducible if and only if the principal ideal (a) is maximal in the set {($b$) where $b$ a nonzero nonunit in R}.

Answer 1

If $(a)$ is not maximal in the given set, there is $b$, a nonzero nonunit, such that $(a)\subsetneq (b)$, which means $a=rb$ for some $r\in R$. So $a$ is reducible.

The converse is equally straight forward. I leave it to you.