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.

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

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es