Definition:Local Ring

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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 $\left({R, +, \circ}\right)$ 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 $1 + \left({- a}\right)$ is 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