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This equivalence (Theorem 3) holds because maximality of {0} forces every nonzero principal ideal to be the entire ring. If {0} is maximal, then for any nonzero a, the ideal (a) strictly contains {0}, so (a) must equal R.

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

This equivalence (Theorem 3) holds because maximality of {0} forces every nonzero principal ideal to be the entire ring. If {0} is maximal, then for any nonzero a, the ideal (a) strictly contains {0}, so (a) must equal R.

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