Characterisation of Local Rings

Theorem
Let $R$ be a ring

Let $\mathfrak m \lhd R$ be a maximal ideal.

1. If the set $R \setminus \mathfrak m$ is precisely the Group of Units $R^\times$ of $R$, then $(R,\mathfrak m)$ is a local ring

2. If $1 + x$ is a unit in $R$ for all $x \in \mathfrak m$ then $(R,\mathfrak m)$ is local.

Proof
1. Suppose that $\mathfrak m$ is the set of non-units of $R$.

Then by Ideal of Unit is Whole Ring, every ideal not equal to $R$ is contained in $\mathfrak m$.

Therefore $\mathfrak m$ is the unique maximal ideal of $R$, so $(R,\mathfrak m)$ is local.

2. Let $x \in R \backslash \mathfrak m$.

Since $\mathfrak m$ is maximal, $x$ and $\mathfrak m$ generate all of $R$.

Therefore $tx + m = 1$ for some $m \in \mathfrak m$, $t \in R$.

Thus $tx = 1 - m \in 1 + \mathfrak m$ is a unit by hypothesis.

Therefore $x$ is a unit. Now use part 1.