Why is the zero ideal {0} not a maximal ideal of the integer ring 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. Recall the definition of a maximal ideal: M is a proper ideal such that no ideal other than R contains M.
2. Observe that every ideal of Z contains the integer 0.
3. Conclude that every ideal of Z contains the zero ideal {0}.
4. Note that there exist larger ideals in Z (such as 2Z or Z itself) that strictly contain {0}.
5. Apply Theorem 3: the zero ideal is maximal if and only if the ring is a field.
6. Verify that Z is not a field because nonzero elements like 2 lack multiplicative inverses.
7. Conclude that {0} is not a maximal ideal of Z.

Example

The video states: 'The zero ideal {0} is not a maximal ideal of Z. The reason is that every ideal of Z contains 0, and therefore also contains {0}; thus {0} can be contained in a larger ideal, violating maximality.' It further notes: 'Because Z is not a field, its zero ideal is not a maximal ideal.'

Common misconceptions

• Assuming that because the zero ideal is maximal in a field, it must be maximal in any ring.
• Confusing the zero ideal with the unit ideal; the unit ideal is not proper, while the zero ideal is proper but not maximal in Z.

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