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}.
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.
Conditions: The ambient ring F is a field.; The ideal under consideration is the zero ideal {0}.
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.
Conditions: The ambient ring F is a field.; The ideal under consideration is the zero ideal {0}.