Definition:Local Ring/Noncommutative

Definition
Let $\left({R, +, \circ}\right)$ be a ring with unity.

Definition 4
$R$ is a local ring 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 $1 + \left({- a}\right)$ 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).