Definition:Local Ring/Commutative

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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.


Also denoted as

One also writes $(A, \mathfrak m)$ for a commutative local ring $A$ with maximal ideal $\mathfrak m$.


Also see


Generalization