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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
    baZ    abb \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}
Last modified 5mo ago
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