What would you like to understand?

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 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.

A quadratic polynomial over a field is irreducible if and only if it has no roots in that field. You can check this by evaluating the polynomial at all elements of the field.

Conditions: The polynomial is of degree 2.; The coefficients are in a field.

A quadratic polynomial over a field is irreducible if and only if it has no roots in that field. You can check this by evaluating the polynomial at all elements of the field.

Conditions: The polynomial is of degree 2.; The coefficients are in a field.

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}.