# Definition:Local Ring/Noncommutative

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## Definition

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 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 $1 + \paren {-a}$ is a unit.

- If the summation $\displaystyle \sum_{i \mathop = 1}^n a_i$ is a unit, then some of the $a_i$ are also units (in particular the empty sum is not a unit).

## Sources

- 1991: T.Y. Lam:
*A First Course in Noncommutative Rings*: Chapter $7$: Local Rings, Semilocal Rings, and Idempotents: $\S 19$: Local Rings