## A ring is Local if and only if the set of Non-Units is an Ideal

## Problem 526

A ring is called **local** if it has a unique maximal ideal.

**(a)** Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$.

**(b)** Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$.

Prove that if every element of $1+M$ is a unit, then $R$ is a local ring.