Why is the unit ideal (1) = Z excluded from being a prime or maximal ideal?

Conditions

• The ambient ring is the ring of integers Z.
• The ideal under consideration is the unit ideal (1) = Z.

Reasoning, step by step

1. Recall the definition of a prime ideal: I must be a proper ideal of R, and ab ∈ I implies $a \in I$ or $b \in I$.
2. Recall the definition of a maximal ideal: M must be a proper ideal of R, and no ideal other than R contains M.
3. Identify that the unit ideal (1) generates the entire ring Z, so (1) = Z.
4. Observe that Z is not a proper subset of itself ($Z = Z$).
5. Conclude that because the unit ideal is not a proper ideal, it is automatically excluded from being a prime or maximal ideal.

Example

The video explicitly notes: 'When discussing maximal ideals, the unit ideal is excluded because it is not a proper ideal.' It also states for prime ideals: 'When $m = 1$, the ideal mZ equals Z itself and is therefore not a prime ideal of Z, since a prime ideal must be a proper ideal.'

Common misconceptions

• Assuming that because Z is an ideal, it could be a maximal ideal, forgetting the 'proper' requirement.
• Thinking that the unit ideal fails the product condition for prime ideals, when it actually fails the foundational proper ideal requirement.

Watch the explanation

Explore next

• Why is the zero ideal {0} a prime ideal of Z?
• What is the definition of a proper ideal?
• Why is mZ a prime ideal when m is prime?