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.
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.
Also see
- Results about local rings can be found here.