Left ideal

Summary

If you have a Noncommutative Ring, then a left ideal is the analogue to the left Nebenklasse.

Definition

Let $R$ be a (non-commutative) Ring. A left-ideal $I$ is a set of numbers that fulfils:

1. $(I,+)$ is a subgroup of $(R,+)$, i.e. closed under inverse and addition
2. For every $r\in R$ we have $rI\in I$

Properties

Distributive property

Ideals fulfill the Distributivgesetz $I(J+K)=IJ+IK$