Every ideal of Z is a principal ideal
Dependencies:
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Principal ideal
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Integer Division Theorem
Let be an ideal of .
Then is a principal ideal, i.e. is of the form for some .
Proof
Let be the smallest positive element of .
Let .
By the integer division theorem, , where .
by absorption and commutativity.
Since , and is the smallest positive element of , .
Therefore, every element of is a multiple of .
Also, all multiples of are in by the absorption property.
Therefore, .
Dependency for: None
Info:
- Depth: 6
- Number of transitive dependencies: 11
Transitive dependencies:
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Group
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Ring
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Ideal
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Identity of a group is unique
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Subgroup
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Inverse of a group element is unique
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Conditions for a subset to be a subgroup
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Condition for a subset to be a subgroup
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Conditions for a subset of a ring to be a subring
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Principal ideal
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Integer Division Theorem