Definition:Local Ring/Commutative

Definition

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 $A$.

Also denoted as

One also writes $\struct {A, \mathfrak m}$ for a commutative local ring $A$ with maximal ideal $\mathfrak m$.

Also see

• Equivalence of Definitions of Commutative Local Ring

• Results about local rings can be found here.

Generalization

• Definition:Noncommutative Local Ring