Why is the zero ideal a maximal ideal of a field?
The zero ideal is a maximal ideal of a field because a field has no nontrivial ideals. The only ideals of a field F are the zero ideal {0} and the unit ideal F itself. Since the only ideals containing {0} are {0} and F, there is no ideal strictly between {0} and F. This perfectly satisfies the definition of a maximal ideal.
Conditions
- The ambient ring F is a field.
- The ideal under consideration is the zero ideal {0}.
Reasoning, step by step
- Recall Theorem 2: A field has no nontrivial ideals; its only ideals are {0} and F.
- State the definition of a maximal ideal: M is a proper ideal such that no ideal other than R contains M.
- Apply this to the zero ideal in a field: the ideals containing {0} are only {0} and F.
- Observe that {0} is a proper ideal of F (since F has nonzero elements).
- Conclude that because there is no intermediate ideal, {0} is a maximal ideal of F.
Example
The video states: 'Corollary: The zero ideal is a maximal ideal of a field.' The voiceover explains: 'Since a field has only the zero ideal and the unit ideal, its zero ideal must be a maximal ideal.'
Common misconceptions
- Assuming that the zero ideal is maximal in any ring, forgetting that this property specifically characterizes fields (as per Theorem 3).
- Confusing the zero ideal being maximal in a field with it being maximal in Z, where it is not.
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