Local ring

We say that a ring $R$ is local if it has a unique maximal ideal $\mathfrak{m}$.
The elements of $R-\mathcal{m}$ are all invertible. If $a\notin \mathfrak{m}$ were not invertible we could consider the ideal generated by $\mathfrak{m}\cup \{a\}$...

Example: ring of germs of functions

Consider continuous functions defined in a real interval containing 0. We can consider the ring ${\mathcal{O}}_0$ formed by class of pairs $(U,f)$ with $0\in U$ and $f:U\mapsto \mathbb{R}$ a continuous function. We establish that two pairs are equivalents if their restricted functions coincide in the intersection. We call this ring the germs of real-valued continuous functions.
The maximal ideal of the ring ${\mathcal{O}}_0$ is

Proposition
A ring $R$ is local iff for every $x\in R$ then $x$ is a unity or $1-x$ is a unity.
$◼$

A source of local ring is the localization of a ring.