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 $\displaystyle \sum_{i=1}^n a_i$ 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.