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.


Sources