# Definition:Local Ring

## 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.

### Definition 3

Let $M \subseteq A$ be the subset of the non-units of $A$.

The ring $A$ is local if and only if $M$ is a proper ideal of $A$.

## 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:

$R$ is nontrivial
the sum of any two non-units of $R$ is a non-unit of $R$.

## 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.