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Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 3 / 10E / a / i / Solution

Suppose

I

is prime. If

(a + I) (b + I) = 0 + I,

then

ab \in I

, so

a \in I

or

b \in I

. Thus one factor is zero and

R / I

is an integral domain.

Conversely, if

R / I

is an integral domain and

ab \in I

, then

(a + I) (b + I) = 0 + I,

so

a + I = 0 + I

or

b + I = 0 + I

. Therefore

a \in I

or

b \in I

, and

I

is prime. This proves the prime ideal quotient criterion.

Solved by gpt-5.6-sol high.

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