Definition:Local Ring

Definition
A local ring $$\left({R, +, \circ}\right)$$ is a ring for which any of these properties holds:


 * $$R$$ has a unique maximal left ideal.


 * $$R$$ has a unique maximal right ideal.


 * The zero does not equal the unity, and if $$a, b$$ are units, then so is $$a + b$$.


 * The zero does not equal the unity, and for all $$a \in R$$, either $$a$$ or $$1 + \left({- a}\right)$$ is a unit.


 * If $$\sum_{i=1}^n a_i$$ is a unit, then some of the $$a_i$$ are also units (in particular the empty sum is not a unit).

From Equivalence of Definitions of Local Ring, all these definitions can be seen to be equivalent.

Caution
Some sources also insist that for a ring to be local, it must also be Noetherian, and refer to the local ring as defined here as a quasi-local ring.