Why is the zero ideal maximal in a unital commutative ring if and only if the ring is a field?

Conditions

• The ambient ring R is a unital commutative ring.
• The ideal under consideration is the zero ideal {0}.

Reasoning, step by step

1. Prove the forward direction (⇒): Assume {0} is a maximal ideal of R.
2. Take any nonzero element $a \in R$. The principal ideal (a) strictly contains {0}.
3. By maximality of {0}, (a) must equal the entire ring R.
4. Since (a) = R, the multiplicative identity $e \in (a)$, meaning there exists $b \in R$ such that ab = e.
5. Conclude that every nonzero a has an inverse b, so R is a field.
6. Prove the reverse direction (⇐): Assume R is a field.
7. By Theorem 2, a field has only two ideals: {0} and R.
8. Since the only ideals containing {0} are {0} and R, {0} is a maximal ideal.

Example

The video states Theorem 3: 'The zero ideal is a maximal ideal of a unital commutative ring if and only if the ring is a field.' The proof shows that if {0} is maximal, (a) = R for any nonzero a, implying ab = e and thus R is a field.

Common misconceptions

• Assuming this theorem applies to non-commutative rings or rings without unity, where the equivalence may fail.
• Confusing the condition that {0} is maximal with the condition that {0} is prime; {0} is prime if and only if the ring is an integral domain.

Watch the explanation

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