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The unit ideal (1) = Z is excluded from being a prime or maximal ideal because both definitions require the ideal to be a proper ideal. A proper ideal must be a strict subset of the ring ($I \ne R$).

Conditions: The ambient ring is the ring of integers Z.; The ideal under consideration is the unit ideal (1) = Z.

The unit ideal (1) = Z is excluded from being a prime or maximal ideal because both definitions require the ideal to be a proper ideal. A proper ideal must be a strict subset of the ring ($I \ne R$).

Conditions: The ambient ring is the ring of integers Z.; The ideal under consideration is the unit ideal (1) = Z.

6Z is not a prime ideal of Z because 6 is a composite number. The definition of a prime ideal requires that if a product ab lies in the ideal, then at least one factor (a or b) must lie in the ideal.

Conditions: The ambient ring is the ring of integers Z.; The ideal under consideration is 6Z.; 6 is a composite number.

6Z is not a prime ideal of Z because 6 is a composite number. The definition of a prime ideal requires that if a product ab lies in the ideal, then at least one factor (a or b) must lie in the ideal.

Conditions: The ambient ring is the ring of integers Z.; The ideal under consideration is 6Z.; 6 is a composite number.

The zero ideal {0} is a prime ideal of Z because it satisfies the definition of a prime ideal. First, {0} is a proper ideal of Z (since {0} ≠ Z).

Conditions: The ambient ring is the ring of integers Z.; The ideal under consideration is the zero ideal {0}.

The zero ideal {0} is a prime ideal of Z because it satisfies the definition of a prime ideal. First, {0} is a proper ideal of Z (since {0} ≠ Z).

Conditions: The ambient ring is the ring of integers Z.; The ideal under consideration is the zero ideal {0}.

The zero ideal {0} is not a maximal ideal of Z because every ideal of Z contains the integer 0, and therefore contains {0}. This means {0} can be properly contained in a larger ideal, violating the definition of a maximal ideal.

Conditions: The ambient ring is the ring of integers Z.; The ideal under consideration is the zero ideal {0}.

The zero ideal {0} is not a maximal ideal of Z because every ideal of Z contains the integer 0, and therefore contains {0}. This means {0} can be properly contained in a larger ideal, violating the definition of a maximal ideal.

Conditions: The ambient ring is the ring of integers Z.; 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}.