My question is how would I go about proving this?
Prove that $R$ is a local ring if and only if all elements of R that are not units form an ideal.
I understand that I need to prove both directions so:
$(\Rightarrow)$ Local ring means has a unique maximal ideal, so I want to show that this implies the elements are not units.
$(\Leftarrow)$ non unit elements of $R$ form an ideal, so if I show this is unique maximal ideal I can then conclude local ring?
Any hints would be appreciated.
The book “Introduction to Commutative Algebra” by Atiyah-Macdonald has in Corollary 1.5:
Every non-unit of $R$ is contained in a maximal ideal.
Let $R$ be local. If $m$ is the only maximal of $R$, then $m$ will be the set of non-units.
Let $n$ be the set of non-units of $R$. Let $m$ be a proper ideal s.t. $n\subseteq m$. Since $m$ is proper, no element of $m$ would be unit and so $m\subseteq n$. So $n$ is maximal.
This is also lemma 3.13 of the book “Steps in Commutative Algebra” by “Sharp” (in that terminology quasi-local means your “local”.