$a\in R\setminus\left\{ 0\right\}$ is irreducible iff $Ra$ is a maximal-principal-ideal (maximal among principal ideals)

Question

Let $R$ be an integral domain. Prove that the element $a\in R\setminus\left\{ 0\right\}$ is irreducible iff $Ra$ is a maximal-principal-ideal (maximal among principal ideals)

Answer 1

If $a$ is reducible, we can write $a=bc$ where neither $b$ nor $c$ are units, and so $Ra \subset Rb\subset R$ meaning that $Ra$ is not maximal-principal. Conversely, if $Ra$ is not maximal-principal, so there is a non-unit $b$ such that $Ra \subset Rb\subset R,$ then $a\in Rb$ so $a=bc$ for some $c\in R$. Note $c$ must be a non-unit since otherwise $Ra=Rb$, and $b$ is a non-unit, so $a$ is reducible.