# 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.

## Also denoted as

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