Ideal

An Additive Subgroup ($I,+$) of a Ring where:

• $\forall r\in R,\exists x\in I,rx\in I$

Showing Ideals

1. Show non-empty
2. Show that $\forall r\in R,\exists x\in I,rx\in I$

Concepts

• Generator for Ideal
• Trivial Ideal
• Generator for Ideal Intersection
• Generator for Ideal Sum
• Principle Ideal

Examples

Showing Even Integers are a Ideal

• Let $x\in \{\text{even numbers}\},x=2k,k\in \mathbb{Z}$
• Let $r\in \mathbb{Z},\text{ then },rx=r(2k)=2(rk)$
• Thus, $rx\in \{\text{even numbers}\}$