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:

$\forall a, b \in \mathbb{Z}$ we have $b \in a\mathbb{Z} \iff a | b$
$I_1 + I_2 = \{a_1 + a_2 \ : \ a_1 \in I_1 , a_2 \in I_2\}$ 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 $d = \gcd(a, b) \Rightarrow a\mathbb{Z} + b\mathbb{Z} = d\mathbb{Z}$

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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:

$\forall a, b \in \mathbb{Z}$ we have $b \in a\mathbb{Z} \iff a | b$

$I_1 + I_2 = \{a_1 + a_2 \ : \ a_1 \in I_1 , a_2 \in I_2\}$ 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 $d = \gcd(a, b) \Rightarrow a\mathbb{Z} + b\mathbb{Z} = d\mathbb{Z}$