# Definition:Local Ring

## Definition

## Commutative ring

Let $A$ be a commutative ring with unity.

### Definition 1

The ring $A$ is **local** if and only if it has a unique maximal ideal.

### Definition 2

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

## Noncommutative ring

Let $\struct {R, +, \circ}$ be a ring with unity.

### Definition 1

$R$ is a **local ring** if and only if it has a unique maximal left ideal.

### Definition 2

$R$ is a **local ring** if and only if it has a unique maximal right ideal.

### Definition 3

Let $\operatorname {rad} R$ be its Jacobson radical.

Then $R$ is a **local ring** if and only if the quotient ring $R / \operatorname{rad} R$ is a division ring.

### Definition 4

$R$ is a **local ring** if and only if 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 $1 + \paren {-a}$ is a unit.

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

## Also defined as

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

- Definition:Local Ring Homomorphism
- Definition:Category of Local Rings
- Results about
**local rings**can be found here.