Why is the zero ideal {0} a prime ideal of Z?

Conditions

• The ambient ring is the ring of integers Z.
• The ideal under consideration is the zero ideal {0}.

Reasoning, step by step

1. Verify that {0} is a proper ideal of Z, as it does not equal Z.
2. State the prime ideal condition: for all a, $b \in Z$, if ab ∈ {0}, then $a \in {0}$ or $b \in {0}$.
3. Translate ab ∈ {0} to the equation ab = 0.
4. Apply the property that the integers have no zero divisors: ab = 0 implies $a = 0$ or $b = 0$.
5. Translate $a = 0$ or $b = 0$ back to ideal membership: $a \in {0}$ or $b \in {0}$.
6. Conclude that {0} satisfies the definition of a prime ideal.

Example

The video states: 'Example: The zero ideal {0} = 0Z is a prime ideal of Z: for a, $b \in Z$, if ab = $0 \in {0}$, then $a = 0 \in {0}$ or $b = 0 \in {0}$.' The voiceover explains: 'The zero ideal is also a prime ideal of Z, because if the product of two integers is zero, then at least one of them must be zero.'

Common misconceptions

• Assuming that only ideals generated by prime numbers can be prime ideals, forgetting that the zero ideal is also a prime ideal in an integral domain like Z.
• Confusing the zero ideal being prime with it being maximal; {0} is prime in Z but not maximal.

Watch the explanation

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