1
$\begingroup$

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)

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.