Definition:Local Ring/Noncommutative

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:

$R$ is nontrivial

the sum of any two non-units of $R$ is a non-unit of $R$.

Also see

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

Also see

• Results about local rings can be found here.

Sources

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