Ideals

Example: Ideals of the integers

Definition - Ideal of Z\mathbb{Z}

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

a+bI and azIa + b \in I \text{ and } az \in I

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

Remarks:

  1. a,bZ\forall a, b \in \mathbb{Z}we have baZabb \in a\mathbb{Z} \iff a | b

  2. I1+I2={a1+a2 : a1I1,a2I2}I_1 + I_2 = \{a_1 + a_2 \ : \ a_1 \in I_1 , a_2 \in I_2\} is an ideal

Example: Consider 18Z+12Z18\mathbb{Z} + 12\mathbb{Z}. This ideal contains 6=181+12(1)18Z+12Z=6Z6 = 18 \cdot 1 + 12 \cdot (-1) \Rightarrow 18\mathbb{Z} + 12\mathbb{Z} = 6\mathbb{Z}

Greatest common divisor

Let a,bZa, b \in \mathbb{Z} be 2 integers. If d=gcd(a,b)aZ+bZ=dZd = \gcd(a, b) \Rightarrow a\mathbb{Z} + b\mathbb{Z} = d\mathbb{Z}