# Ideals

## Example: Ideals of the integers 

**Definition - Ideal of** $$\mathbb{Z}$$

$$I \subseteq \mathbb{Z}$$is an ideal $$\iff \forall \ a, b \in I \text{ and} , z\ \in \mathbb{Z}$$we have 

$$a + b \in I \text{ and } az \in I$$

**Example**: $$a\mathbb{Z} = {az \ : \ z \in \mathbb{Z} } \to 2\mathbb{Z}, 3\mathbb{Z}, 4\mathbb{Z}, \dots$$  - multiples of $$a$$

**Remarks**: 

1. $$\forall a, b \in \mathbb{Z}$$we have $$b \in a\mathbb{Z} \iff a | b$$
2. $$I\_1 + I\_2 = {a\_1 + a\_2 \ : \ a\_1 \in I\_1 , a\_2 \in I\_2}$$ is an ideal

**Example**: Consider $$18\mathbb{Z} + 12\mathbb{Z}$$. This ideal contains $$6 = 18 \cdot 1 + 12 \cdot (-1) \Rightarrow 18\mathbb{Z} + 12\mathbb{Z} = 6\mathbb{Z}$$

**Greatest common divisor**

> Let $$a, b \in \mathbb{Z}$$ be 2 integers. If $$d = \gcd(a, b) \Rightarrow a\mathbb{Z} + b\mathbb{Z} = d\mathbb{Z}$$