Definition:Local Ring

Commutative ring
Let $A$ be a commutative ring with unity.

Definition 1
The ring $A$ is local it has a unique maximal ideal.

Definition 2
The ring $A$ is local it is nontrivial and the sum of any two non-units is a non-unit.

Non-commutative ring
Let $\left({R, +, \circ}\right)$ be a ring with unity.

Definition 1
$R$ is a local ring if it has a unique maximal left ideal.

Definition 2
$R$ is a local ring if it has a unique maximal right ideal.

Definition 3
$R$ is a local ring it is nontrivial and the sum of any two non-units is a non-unit.


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


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

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.

Also see

 * Equivalence of Definitions of Local Ring
 * Definition:Local Ring Homomorphism
 * Definition:Category of Local Rings